Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 75.9% → 76.6%

Time: 40.5s

Alternatives: 16

Speedup: 0.3×

Specification

\[\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land maxAniso = 16\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := t\_1 \cdot t\_1 + t\_4 \cdot t\_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;t\_6 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_4 (* (floor h) dY.v))
        (t_5 (+ (* t_1 t_1) (* t_4 t_4)))
        (t_6 (/ 1.0 (sqrt (fmax t_3 t_5)))))
   (if (>= t_3 t_5) (* t_6 t_2) (* t_6 t_1))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = (t_2 * t_2) + (t_0 * t_0);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = (t_1 * t_1) + (t_4 * t_4);
	float t_6 = 1.0f / sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_3 >= t_5) {
		tmp = t_6 * t_2;
	} else {
		tmp = t_6 * t_1;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(fmax(t_3, t_5)))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Initial Program: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := t\_1 \cdot t\_1 + t\_4 \cdot t\_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_3, t\_5\right)}}\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;t\_6 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_4 (* (floor h) dY.v))
        (t_5 (+ (* t_1 t_1) (* t_4 t_4)))
        (t_6 (/ 1.0 (sqrt (fmax t_3 t_5)))))
   (if (>= t_3 t_5) (* t_6 t_2) (* t_6 t_1))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = (t_2 * t_2) + (t_0 * t_0);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = (t_1 * t_1) + (t_4 * t_4);
	float t_6 = 1.0f / sqrtf(fmaxf(t_3, t_5));
	float tmp;
	if (t_3 >= t_5) {
		tmp = t_6 * t_2;
	} else {
		tmp = t_6 * t_1;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(fmax(t_3, t_5)))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Alternative 1: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_5 := \left(t\_4 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_6 := \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \\ t_7 := t\_1 \cdot t\_1 + t\_0 \cdot t\_0\\ t_8 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_9 := t\_3 \cdot t\_3 + t\_8 \cdot t\_8\\ t_10 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_11 := \frac{1}{\sqrt{\mathsf{max}\left(t\_7, t\_9\right)}}\\ t_12 := \begin{array}{l} \mathbf{if}\;t\_7 \geq t\_9:\\ \;\;\;\;t\_11 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_11 \cdot t\_3\\ \end{array}\\ t_13 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_14 := \mathsf{fma}\left(dY.v \cdot dY.v, t\_10, \left(t\_13 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ t_15 := \sqrt{\mathsf{max}\left(t\_5, t\_14\right)}\\ t_16 := t\_2 \cdot \left(dY.u \cdot dY.u\right)\\ t_17 := \sqrt{\mathsf{max}\left(t\_5, t\_16\right)}\\ \mathbf{if}\;t\_12 \leq -0.9959999918937683:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_14:\\ \;\;\;\;\frac{t\_4}{t\_15}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_15}\\ \end{array}\\ \mathbf{elif}\;t\_12 \leq 0.9999949932098389:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_7 \geq t\_6:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_7, t\_6\right)}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_10 \cdot dX.v, dX.v, t\_2 \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(t\_2 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_10\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array}\\ \mathbf{elif}\;t\_5 \geq t\_16:\\ \;\;\;\;\frac{t\_4}{t\_17}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_17}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) (floor w)))
        (t_3 (* (floor w) dY.u))
        (t_4 (* dX.u (floor w)))
        (t_5 (* (* t_4 dX.u) (floor w)))
        (t_6 (* (* (* dY.v (floor h)) dY.v) (floor h)))
        (t_7 (+ (* t_1 t_1) (* t_0 t_0)))
        (t_8 (* (floor h) dY.v))
        (t_9 (+ (* t_3 t_3) (* t_8 t_8)))
        (t_10 (* (floor h) (floor h)))
        (t_11 (/ 1.0 (sqrt (fmax t_7 t_9))))
        (t_12 (if (>= t_7 t_9) (* t_11 t_1) (* t_11 t_3)))
        (t_13 (* dY.u (floor w)))
        (t_14 (fma (* dY.v dY.v) t_10 (* (* t_13 dY.u) (floor w))))
        (t_15 (sqrt (fmax t_5 t_14)))
        (t_16 (* t_2 (* dY.u dY.u)))
        (t_17 (sqrt (fmax t_5 t_16))))
   (if (<= t_12 -0.9959999918937683)
     (if (>= t_5 t_14) (/ t_4 t_15) (/ t_13 t_15))
     (if (<= t_12 0.9999949932098389)
       (if (>= t_7 t_6)
         (* (/ 1.0 (sqrt (fmax t_7 t_6))) t_1)
         (fma
          (/
           dY.u
           (sqrt
            (fmax
             (fma (* t_10 dX.v) dX.v (* t_2 (* dX.u dX.u)))
             (fma (* t_2 dY.u) dY.u (* (* dY.v dY.v) t_10)))))
          (floor w)
          0.0))
       (if (>= t_5 t_16) (/ t_4 t_17) (/ t_13 t_17))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * floorf(w);
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = dX_46_u * floorf(w);
	float t_5 = (t_4 * dX_46_u) * floorf(w);
	float t_6 = ((dY_46_v * floorf(h)) * dY_46_v) * floorf(h);
	float t_7 = (t_1 * t_1) + (t_0 * t_0);
	float t_8 = floorf(h) * dY_46_v;
	float t_9 = (t_3 * t_3) + (t_8 * t_8);
	float t_10 = floorf(h) * floorf(h);
	float t_11 = 1.0f / sqrtf(fmaxf(t_7, t_9));
	float tmp;
	if (t_7 >= t_9) {
		tmp = t_11 * t_1;
	} else {
		tmp = t_11 * t_3;
	}
	float t_12 = tmp;
	float t_13 = dY_46_u * floorf(w);
	float t_14 = fmaf((dY_46_v * dY_46_v), t_10, ((t_13 * dY_46_u) * floorf(w)));
	float t_15 = sqrtf(fmaxf(t_5, t_14));
	float t_16 = t_2 * (dY_46_u * dY_46_u);
	float t_17 = sqrtf(fmaxf(t_5, t_16));
	float tmp_2;
	if (t_12 <= -0.9959999918937683f) {
		float tmp_3;
		if (t_5 >= t_14) {
			tmp_3 = t_4 / t_15;
		} else {
			tmp_3 = t_13 / t_15;
		}
		tmp_2 = tmp_3;
	} else if (t_12 <= 0.9999949932098389f) {
		float tmp_4;
		if (t_7 >= t_6) {
			tmp_4 = (1.0f / sqrtf(fmaxf(t_7, t_6))) * t_1;
		} else {
			tmp_4 = fmaf((dY_46_u / sqrtf(fmaxf(fmaf((t_10 * dX_46_v), dX_46_v, (t_2 * (dX_46_u * dX_46_u))), fmaf((t_2 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_10))))), floorf(w), 0.0f);
		}
		tmp_2 = tmp_4;
	} else if (t_5 >= t_16) {
		tmp_2 = t_4 / t_17;
	} else {
		tmp_2 = t_13 / t_17;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * floor(w))
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = Float32(Float32(t_4 * dX_46_u) * floor(w))
	t_6 = Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))
	t_7 = Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0))
	t_8 = Float32(floor(h) * dY_46_v)
	t_9 = Float32(Float32(t_3 * t_3) + Float32(t_8 * t_8))
	t_10 = Float32(floor(h) * floor(h))
	t_11 = Float32(Float32(1.0) / sqrt(fmax(t_7, t_9)))
	tmp = Float32(0.0)
	if (t_7 >= t_9)
		tmp = Float32(t_11 * t_1);
	else
		tmp = Float32(t_11 * t_3);
	end
	t_12 = tmp
	t_13 = Float32(dY_46_u * floor(w))
	t_14 = fma(Float32(dY_46_v * dY_46_v), t_10, Float32(Float32(t_13 * dY_46_u) * floor(w)))
	t_15 = sqrt(fmax(t_5, t_14))
	t_16 = Float32(t_2 * Float32(dY_46_u * dY_46_u))
	t_17 = sqrt(fmax(t_5, t_16))
	tmp_2 = Float32(0.0)
	if (t_12 <= Float32(-0.9959999918937683))
		tmp_3 = Float32(0.0)
		if (t_5 >= t_14)
			tmp_3 = Float32(t_4 / t_15);
		else
			tmp_3 = Float32(t_13 / t_15);
		end
		tmp_2 = tmp_3;
	elseif (t_12 <= Float32(0.9999949932098389))
		tmp_4 = Float32(0.0)
		if (t_7 >= t_6)
			tmp_4 = Float32(Float32(Float32(1.0) / sqrt(fmax(t_7, t_6))) * t_1);
		else
			tmp_4 = fma(Float32(dY_46_u / sqrt(fmax(fma(Float32(t_10 * dX_46_v), dX_46_v, Float32(t_2 * Float32(dX_46_u * dX_46_u))), fma(Float32(t_2 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_10))))), floor(w), Float32(0.0));
		end
		tmp_2 = tmp_4;
	elseif (t_5 >= t_16)
		tmp_2 = Float32(t_4 / t_17);
	else
		tmp_2 = Float32(t_13 / t_17);
	end
	return tmp_2
end

Derivation

1. Split input into 3 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.995999992

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 92.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 91.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 94.7%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 94.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 98.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 98.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   if -0.995999992 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.999994993

   1. Initial program 63.3%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 63.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   4. Taylor expanded in dY.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   6. Applied rewrites 63.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   7. Taylor expanded in dY.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   9. Applied rewrites 64.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   if 0.999994993 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.5%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 93.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 93.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 95.0%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   20. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 95.0

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 95.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   23. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 99.4

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 99.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor \color{blue}{w}\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   26. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 99.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   27. Applied rewrites 99.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor \color{blue}{w}\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := \left(t\_3 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_5 := \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \\ t_6 := t\_1 \cdot t\_1 + t\_0 \cdot t\_0\\ t_7 := \frac{1}{\sqrt{\mathsf{max}\left(t\_6, t\_5\right)}}\\ t_8 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_9 := t\_2 \cdot t\_2 + t\_8 \cdot t\_8\\ t_10 := \frac{1}{\sqrt{\mathsf{max}\left(t\_6, t\_9\right)}}\\ t_11 := \begin{array}{l} \mathbf{if}\;t\_6 \geq t\_9:\\ \;\;\;\;t\_10 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_10 \cdot t\_2\\ \end{array}\\ t_12 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_13 := \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(t\_12 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ t_14 := \sqrt{\mathsf{max}\left(t\_4, t\_13\right)}\\ t_15 := \begin{array}{l} \mathbf{if}\;t\_4 \geq t\_13:\\ \;\;\;\;\frac{t\_3}{t\_14}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_12}{t\_14}\\ \end{array}\\ \mathbf{if}\;t\_11 \leq -0.9959999918937683:\\ \;\;\;\;t\_15\\ \mathbf{elif}\;t\_11 \leq 0.15000000596046448:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_6 \geq t\_5:\\ \;\;\;\;t\_7 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_7 \cdot t\_2\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_15\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u))
        (t_3 (* dX.u (floor w)))
        (t_4 (* (* t_3 dX.u) (floor w)))
        (t_5 (* (* (* dY.v (floor h)) dY.v) (floor h)))
        (t_6 (+ (* t_1 t_1) (* t_0 t_0)))
        (t_7 (/ 1.0 (sqrt (fmax t_6 t_5))))
        (t_8 (* (floor h) dY.v))
        (t_9 (+ (* t_2 t_2) (* t_8 t_8)))
        (t_10 (/ 1.0 (sqrt (fmax t_6 t_9))))
        (t_11 (if (>= t_6 t_9) (* t_10 t_1) (* t_10 t_2)))
        (t_12 (* dY.u (floor w)))
        (t_13
         (fma
          (* dY.v dY.v)
          (* (floor h) (floor h))
          (* (* t_12 dY.u) (floor w))))
        (t_14 (sqrt (fmax t_4 t_13)))
        (t_15 (if (>= t_4 t_13) (/ t_3 t_14) (/ t_12 t_14))))
   (if (<= t_11 -0.9959999918937683)
     t_15
     (if (<= t_11 0.15000000596046448)
       (if (>= t_6 t_5) (* t_7 t_1) (* t_7 t_2))
       t_15))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = (t_3 * dX_46_u) * floorf(w);
	float t_5 = ((dY_46_v * floorf(h)) * dY_46_v) * floorf(h);
	float t_6 = (t_1 * t_1) + (t_0 * t_0);
	float t_7 = 1.0f / sqrtf(fmaxf(t_6, t_5));
	float t_8 = floorf(h) * dY_46_v;
	float t_9 = (t_2 * t_2) + (t_8 * t_8);
	float t_10 = 1.0f / sqrtf(fmaxf(t_6, t_9));
	float tmp;
	if (t_6 >= t_9) {
		tmp = t_10 * t_1;
	} else {
		tmp = t_10 * t_2;
	}
	float t_11 = tmp;
	float t_12 = dY_46_u * floorf(w);
	float t_13 = fmaf((dY_46_v * dY_46_v), (floorf(h) * floorf(h)), ((t_12 * dY_46_u) * floorf(w)));
	float t_14 = sqrtf(fmaxf(t_4, t_13));
	float tmp_1;
	if (t_4 >= t_13) {
		tmp_1 = t_3 / t_14;
	} else {
		tmp_1 = t_12 / t_14;
	}
	float t_15 = tmp_1;
	float tmp_2;
	if (t_11 <= -0.9959999918937683f) {
		tmp_2 = t_15;
	} else if (t_11 <= 0.15000000596046448f) {
		float tmp_3;
		if (t_6 >= t_5) {
			tmp_3 = t_7 * t_1;
		} else {
			tmp_3 = t_7 * t_2;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_15;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(Float32(t_3 * dX_46_u) * floor(w))
	t_5 = Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))
	t_6 = Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0))
	t_7 = Float32(Float32(1.0) / sqrt(fmax(t_6, t_5)))
	t_8 = Float32(floor(h) * dY_46_v)
	t_9 = Float32(Float32(t_2 * t_2) + Float32(t_8 * t_8))
	t_10 = Float32(Float32(1.0) / sqrt(fmax(t_6, t_9)))
	tmp = Float32(0.0)
	if (t_6 >= t_9)
		tmp = Float32(t_10 * t_1);
	else
		tmp = Float32(t_10 * t_2);
	end
	t_11 = tmp
	t_12 = Float32(dY_46_u * floor(w))
	t_13 = fma(Float32(dY_46_v * dY_46_v), Float32(floor(h) * floor(h)), Float32(Float32(t_12 * dY_46_u) * floor(w)))
	t_14 = sqrt(fmax(t_4, t_13))
	tmp_1 = Float32(0.0)
	if (t_4 >= t_13)
		tmp_1 = Float32(t_3 / t_14);
	else
		tmp_1 = Float32(t_12 / t_14);
	end
	t_15 = tmp_1
	tmp_2 = Float32(0.0)
	if (t_11 <= Float32(-0.9959999918937683))
		tmp_2 = t_15;
	elseif (t_11 <= Float32(0.15000000596046448))
		tmp_3 = Float32(0.0)
		if (t_6 >= t_5)
			tmp_3 = Float32(t_7 * t_1);
		else
			tmp_3 = Float32(t_7 * t_2);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_15;
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.995999992 or 0.150000006 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 91.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 91.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 93.6%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 93.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 97.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 97.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   if -0.995999992 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.150000006

   1. Initial program 62.2%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 60.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 60.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 51.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dY.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \color{blue}{\left\lfloor h\right\rfloor }\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot \color{blue}{\left\lfloor h\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot \color{blue}{\left\lfloor h\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. associate-*l* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\right) \cdot \left\lfloor \color{blue}{h}\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lift-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      9. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor \color{blue}{h}\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      10. lower-*.f32 52.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor \color{blue}{h}\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      11. lift-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      12. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      13. lower-*.f32 52.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 52.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       2. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       3. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       4. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       6. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\right) \cdot \left\lfloor \color{blue}{h}\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       7. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       8. lift-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       9. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor \color{blue}{h}\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       10. lower-*.f32 54.7

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor \color{blue}{h}\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       11. lift-*.f32 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       12. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       13. lower-*.f32 54.7

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   15. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       2. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       3. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       4. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       6. associate-*l* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       7. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       8. lift-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       9. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       10. lower-*.f32 62.6

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       11. lift-*.f32 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       12. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       13. lower-*.f32 62.6

           \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 62.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := \left(t\_3 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_5 := t\_1 \cdot t\_1 + t\_0 \cdot t\_0\\ t_6 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_7 := t\_2 \cdot t\_2 + t\_6 \cdot t\_6\\ t_8 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_9 := \left(dY.v \cdot dY.v\right) \cdot t\_8\\ t_10 := \frac{1}{\sqrt{\mathsf{max}\left(t\_5, t\_9\right)}}\\ t_11 := \frac{1}{\sqrt{\mathsf{max}\left(t\_5, t\_7\right)}}\\ t_12 := \begin{array}{l} \mathbf{if}\;t\_5 \geq t\_7:\\ \;\;\;\;t\_11 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_11 \cdot t\_2\\ \end{array}\\ t_13 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_14 := \mathsf{fma}\left(dY.v \cdot dY.v, t\_8, \left(t\_13 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ t_15 := \sqrt{\mathsf{max}\left(t\_4, t\_14\right)}\\ t_16 := \begin{array}{l} \mathbf{if}\;t\_4 \geq t\_14:\\ \;\;\;\;\frac{t\_3}{t\_15}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_15}\\ \end{array}\\ \mathbf{if}\;t\_12 \leq -0.9959999918937683:\\ \;\;\;\;t\_16\\ \mathbf{elif}\;t\_12 \leq 0.15000000596046448:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_9:\\ \;\;\;\;t\_10 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_10 \cdot t\_2\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_16\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u))
        (t_3 (* dX.u (floor w)))
        (t_4 (* (* t_3 dX.u) (floor w)))
        (t_5 (+ (* t_1 t_1) (* t_0 t_0)))
        (t_6 (* (floor h) dY.v))
        (t_7 (+ (* t_2 t_2) (* t_6 t_6)))
        (t_8 (* (floor h) (floor h)))
        (t_9 (* (* dY.v dY.v) t_8))
        (t_10 (/ 1.0 (sqrt (fmax t_5 t_9))))
        (t_11 (/ 1.0 (sqrt (fmax t_5 t_7))))
        (t_12 (if (>= t_5 t_7) (* t_11 t_1) (* t_11 t_2)))
        (t_13 (* dY.u (floor w)))
        (t_14 (fma (* dY.v dY.v) t_8 (* (* t_13 dY.u) (floor w))))
        (t_15 (sqrt (fmax t_4 t_14)))
        (t_16 (if (>= t_4 t_14) (/ t_3 t_15) (/ t_13 t_15))))
   (if (<= t_12 -0.9959999918937683)
     t_16
     (if (<= t_12 0.15000000596046448)
       (if (>= t_5 t_9) (* t_10 t_1) (* t_10 t_2))
       t_16))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = (t_3 * dX_46_u) * floorf(w);
	float t_5 = (t_1 * t_1) + (t_0 * t_0);
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = (t_2 * t_2) + (t_6 * t_6);
	float t_8 = floorf(h) * floorf(h);
	float t_9 = (dY_46_v * dY_46_v) * t_8;
	float t_10 = 1.0f / sqrtf(fmaxf(t_5, t_9));
	float t_11 = 1.0f / sqrtf(fmaxf(t_5, t_7));
	float tmp;
	if (t_5 >= t_7) {
		tmp = t_11 * t_1;
	} else {
		tmp = t_11 * t_2;
	}
	float t_12 = tmp;
	float t_13 = dY_46_u * floorf(w);
	float t_14 = fmaf((dY_46_v * dY_46_v), t_8, ((t_13 * dY_46_u) * floorf(w)));
	float t_15 = sqrtf(fmaxf(t_4, t_14));
	float tmp_1;
	if (t_4 >= t_14) {
		tmp_1 = t_3 / t_15;
	} else {
		tmp_1 = t_13 / t_15;
	}
	float t_16 = tmp_1;
	float tmp_2;
	if (t_12 <= -0.9959999918937683f) {
		tmp_2 = t_16;
	} else if (t_12 <= 0.15000000596046448f) {
		float tmp_3;
		if (t_5 >= t_9) {
			tmp_3 = t_10 * t_1;
		} else {
			tmp_3 = t_10 * t_2;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_16;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(Float32(t_3 * dX_46_u) * floor(w))
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_2 * t_2) + Float32(t_6 * t_6))
	t_8 = Float32(floor(h) * floor(h))
	t_9 = Float32(Float32(dY_46_v * dY_46_v) * t_8)
	t_10 = Float32(Float32(1.0) / sqrt(fmax(t_5, t_9)))
	t_11 = Float32(Float32(1.0) / sqrt(fmax(t_5, t_7)))
	tmp = Float32(0.0)
	if (t_5 >= t_7)
		tmp = Float32(t_11 * t_1);
	else
		tmp = Float32(t_11 * t_2);
	end
	t_12 = tmp
	t_13 = Float32(dY_46_u * floor(w))
	t_14 = fma(Float32(dY_46_v * dY_46_v), t_8, Float32(Float32(t_13 * dY_46_u) * floor(w)))
	t_15 = sqrt(fmax(t_4, t_14))
	tmp_1 = Float32(0.0)
	if (t_4 >= t_14)
		tmp_1 = Float32(t_3 / t_15);
	else
		tmp_1 = Float32(t_13 / t_15);
	end
	t_16 = tmp_1
	tmp_2 = Float32(0.0)
	if (t_12 <= Float32(-0.9959999918937683))
		tmp_2 = t_16;
	elseif (t_12 <= Float32(0.15000000596046448))
		tmp_3 = Float32(0.0)
		if (t_5 >= t_9)
			tmp_3 = Float32(t_10 * t_1);
		else
			tmp_3 = Float32(t_10 * t_2);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_16;
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.995999992 or 0.150000006 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 91.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 91.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 93.6%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 93.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 97.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 97.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   if -0.995999992 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.150000006

   1. Initial program 62.2%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dY.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq {dY.v}^{2} \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \color{blue}{\left\lfloor h\right\rfloor }\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \color{blue}{\left\lfloor h\right\rfloor }\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor \color{blue}{h}\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. lift-floor.f32 62.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 62.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dY.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), {dY.v}^{2} \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor \color{blue}{h}\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. lift-floor.f32 63.7

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 63.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dY.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. lift-floor.f32 62.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 62.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\\ t_5 := \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \\ t_6 := \sqrt{\mathsf{max}\left(t\_5, t\_4\right)}\\ t_7 := \mathsf{fma}\left(t\_3 \cdot dX.u, \left\lfloor w\right\rfloor , t\_5\right)\\ t_8 := t\_1 \cdot t\_1 + t\_0 \cdot t\_0\\ t_9 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_10 := t\_2 \cdot t\_2 + t\_9 \cdot t\_9\\ t_11 := \frac{1}{\sqrt{\mathsf{max}\left(t\_8, t\_10\right)}}\\ t_12 := \begin{array}{l} \mathbf{if}\;t\_8 \geq t\_10:\\ \;\;\;\;t\_11 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_11 \cdot t\_2\\ \end{array}\\ t_13 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_14 := \left(t\_13 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ t_15 := \sqrt{\mathsf{max}\left(t\_7, t\_14\right)}\\ t_16 := \begin{array}{l} \mathbf{if}\;t\_7 \geq t\_14:\\ \;\;\;\;\frac{t\_3}{t\_15}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_15}\\ \end{array}\\ \mathbf{if}\;t\_12 \leq -0.0005000000237487257:\\ \;\;\;\;t\_16\\ \mathbf{elif}\;t\_12 \leq 0.6000000238418579:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_4:\\ \;\;\;\;\frac{t\_1}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_6}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_16\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u))
        (t_3 (* dX.u (floor w)))
        (t_4
         (fma
          (* (* (floor h) (floor h)) dY.v)
          dY.v
          (* (* (* dY.u dY.u) (floor w)) (floor w))))
        (t_5 (* (* (* dX.v (floor h)) dX.v) (floor h)))
        (t_6 (sqrt (fmax t_5 t_4)))
        (t_7 (fma (* t_3 dX.u) (floor w) t_5))
        (t_8 (+ (* t_1 t_1) (* t_0 t_0)))
        (t_9 (* (floor h) dY.v))
        (t_10 (+ (* t_2 t_2) (* t_9 t_9)))
        (t_11 (/ 1.0 (sqrt (fmax t_8 t_10))))
        (t_12 (if (>= t_8 t_10) (* t_11 t_1) (* t_11 t_2)))
        (t_13 (* dY.u (floor w)))
        (t_14 (* (* t_13 dY.u) (floor w)))
        (t_15 (sqrt (fmax t_7 t_14)))
        (t_16 (if (>= t_7 t_14) (/ t_3 t_15) (/ t_13 t_15))))
   (if (<= t_12 -0.0005000000237487257)
     t_16
     (if (<= t_12 0.6000000238418579)
       (if (>= t_5 t_4) (/ t_1 t_6) (/ t_2 t_6))
       t_16))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = fmaf(((floorf(h) * floorf(h)) * dY_46_v), dY_46_v, (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)));
	float t_5 = ((dX_46_v * floorf(h)) * dX_46_v) * floorf(h);
	float t_6 = sqrtf(fmaxf(t_5, t_4));
	float t_7 = fmaf((t_3 * dX_46_u), floorf(w), t_5);
	float t_8 = (t_1 * t_1) + (t_0 * t_0);
	float t_9 = floorf(h) * dY_46_v;
	float t_10 = (t_2 * t_2) + (t_9 * t_9);
	float t_11 = 1.0f / sqrtf(fmaxf(t_8, t_10));
	float tmp;
	if (t_8 >= t_10) {
		tmp = t_11 * t_1;
	} else {
		tmp = t_11 * t_2;
	}
	float t_12 = tmp;
	float t_13 = dY_46_u * floorf(w);
	float t_14 = (t_13 * dY_46_u) * floorf(w);
	float t_15 = sqrtf(fmaxf(t_7, t_14));
	float tmp_1;
	if (t_7 >= t_14) {
		tmp_1 = t_3 / t_15;
	} else {
		tmp_1 = t_13 / t_15;
	}
	float t_16 = tmp_1;
	float tmp_2;
	if (t_12 <= -0.0005000000237487257f) {
		tmp_2 = t_16;
	} else if (t_12 <= 0.6000000238418579f) {
		float tmp_3;
		if (t_5 >= t_4) {
			tmp_3 = t_1 / t_6;
		} else {
			tmp_3 = t_2 / t_6;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_16;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = fma(Float32(Float32(floor(h) * floor(h)) * dY_46_v), dY_46_v, Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))
	t_5 = Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))
	t_6 = sqrt(fmax(t_5, t_4))
	t_7 = fma(Float32(t_3 * dX_46_u), floor(w), t_5)
	t_8 = Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0))
	t_9 = Float32(floor(h) * dY_46_v)
	t_10 = Float32(Float32(t_2 * t_2) + Float32(t_9 * t_9))
	t_11 = Float32(Float32(1.0) / sqrt(fmax(t_8, t_10)))
	tmp = Float32(0.0)
	if (t_8 >= t_10)
		tmp = Float32(t_11 * t_1);
	else
		tmp = Float32(t_11 * t_2);
	end
	t_12 = tmp
	t_13 = Float32(dY_46_u * floor(w))
	t_14 = Float32(Float32(t_13 * dY_46_u) * floor(w))
	t_15 = sqrt(fmax(t_7, t_14))
	tmp_1 = Float32(0.0)
	if (t_7 >= t_14)
		tmp_1 = Float32(t_3 / t_15);
	else
		tmp_1 = Float32(t_13 / t_15);
	end
	t_16 = tmp_1
	tmp_2 = Float32(0.0)
	if (t_12 <= Float32(-0.0005000000237487257))
		tmp_2 = t_16;
	elseif (t_12 <= Float32(0.6000000238418579))
		tmp_3 = Float32(0.0)
		if (t_5 >= t_4)
			tmp_3 = Float32(t_1 / t_6);
		else
			tmp_3 = Float32(t_2 / t_6);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_16;
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -5.00000024e-4 or 0.600000024 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.3%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dY.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. pow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. lift-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(dY.u \cdot dY.u\right) \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \color{blue}{\left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \color{blue}{\left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor \color{blue}{w}\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lift-floor.f32 93.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   12. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.u \cdot dY.u\right) \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       3. lift-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.u \cdot dY.u\right) \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       4. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \color{blue}{\left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \color{blue}{\left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       6. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor \color{blue}{w}\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       7. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       8. lift-floor.f32 96.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 96.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   15. Step-by-step derivation

       1. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       3. lift-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       4. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       6. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       7. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

       8. lift-floor.f32 95.1

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 95.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 95.5%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right) \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ } \end{array}} \]

   if -5.00000024e-4 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.600000024

   1. Initial program 60.8%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.8

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 60.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.4

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 60.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 61.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 61.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   12. Applied rewrites 61.7%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ } \end{array}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;t\_1 \cdot t\_1 + t\_0 \cdot t\_0 \geq t\_2 \cdot t\_2 + t\_4 \cdot t\_4:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot t\_3\right), \mathsf{fma}\left(t\_5 \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_5 \cdot dX.v, dX.v, t\_3 \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(t\_3 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_5\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u))
        (t_3 (* (floor w) (floor w)))
        (t_4 (* (floor h) dY.v))
        (t_5 (* (floor h) (floor h))))
   (if (>= (+ (* t_1 t_1) (* t_0 t_0)) (+ (* t_2 t_2) (* t_4 t_4)))
     (/
      (* (- dX.u) (floor w))
      (-
       (sqrt
        (fmax
         (fma (* (* dX.v (floor h)) dX.v) (floor h) (* (* dX.u dX.u) t_3))
         (fma (* t_5 dY.v) dY.v (* (* (* dY.u dY.u) (floor w)) (floor w)))))))
     (fma
      (/
       dY.u
       (sqrt
        (fmax
         (fma (* t_5 dX.v) dX.v (* t_3 (* dX.u dX.u)))
         (fma (* t_3 dY.u) dY.u (* (* dY.v dY.v) t_5)))))
      (floor w)
      0.0))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(w) * floorf(w);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = floorf(h) * floorf(h);
	float tmp;
	if (((t_1 * t_1) + (t_0 * t_0)) >= ((t_2 * t_2) + (t_4 * t_4))) {
		tmp = (-dX_46_u * floorf(w)) / -sqrtf(fmaxf(fmaf(((dX_46_v * floorf(h)) * dX_46_v), floorf(h), ((dX_46_u * dX_46_u) * t_3)), fmaf((t_5 * dY_46_v), dY_46_v, (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)))));
	} else {
		tmp = fmaf((dY_46_u / sqrtf(fmaxf(fmaf((t_5 * dX_46_v), dX_46_v, (t_3 * (dX_46_u * dX_46_u))), fmaf((t_3 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_5))))), floorf(w), 0.0f);
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(w) * floor(w))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0)) >= Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)))
		tmp = Float32(Float32(Float32(-dX_46_u) * floor(w)) / Float32(-sqrt(fmax(fma(Float32(Float32(dX_46_v * floor(h)) * dX_46_v), floor(h), Float32(Float32(dX_46_u * dX_46_u) * t_3)), fma(Float32(t_5 * dY_46_v), dY_46_v, Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))))));
	else
		tmp = fma(Float32(dY_46_u / sqrt(fmax(fma(Float32(t_5 * dX_46_v), dX_46_v, Float32(t_3 * Float32(dX_46_u * dX_46_u))), fma(Float32(t_3 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_5))))), floor(w), Float32(0.0));
	end
	return tmp
end

Derivation

1. Initial program 75.9%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]
6. Add Preprocessing

Alternative 6: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;t\_1 \cdot t\_1 + t\_0 \cdot t\_0 \geq t\_2 \cdot t\_2 + t\_4 \cdot t\_4:\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot t\_3\right), \mathsf{fma}\left(t\_5 \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_5 \cdot dX.v, dX.v, t\_3 \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(t\_3 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_5\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u))
        (t_3 (* (floor w) (floor w)))
        (t_4 (* (floor h) dY.v))
        (t_5 (* (floor h) (floor h))))
   (if (>= (+ (* t_1 t_1) (* t_0 t_0)) (+ (* t_2 t_2) (* t_4 t_4)))
     (/
      (* (- dX.u) (floor w))
      (-
       (sqrt
        (fmax
         (fma (* (* dX.v dX.v) (floor h)) (floor h) (* (* dX.u dX.u) t_3))
         (fma (* t_5 dY.v) dY.v (* (* (* dY.u dY.u) (floor w)) (floor w)))))))
     (fma
      (/
       dY.u
       (sqrt
        (fmax
         (fma (* t_5 dX.v) dX.v (* t_3 (* dX.u dX.u)))
         (fma (* t_3 dY.u) dY.u (* (* dY.v dY.v) t_5)))))
      (floor w)
      0.0))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = floorf(w) * floorf(w);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = floorf(h) * floorf(h);
	float tmp;
	if (((t_1 * t_1) + (t_0 * t_0)) >= ((t_2 * t_2) + (t_4 * t_4))) {
		tmp = (-dX_46_u * floorf(w)) / -sqrtf(fmaxf(fmaf(((dX_46_v * dX_46_v) * floorf(h)), floorf(h), ((dX_46_u * dX_46_u) * t_3)), fmaf((t_5 * dY_46_v), dY_46_v, (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)))));
	} else {
		tmp = fmaf((dY_46_u / sqrtf(fmaxf(fmaf((t_5 * dX_46_v), dX_46_v, (t_3 * (dX_46_u * dX_46_u))), fmaf((t_3 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_5))))), floorf(w), 0.0f);
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(floor(w) * floor(w))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0)) >= Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)))
		tmp = Float32(Float32(Float32(-dX_46_u) * floor(w)) / Float32(-sqrt(fmax(fma(Float32(Float32(dX_46_v * dX_46_v) * floor(h)), floor(h), Float32(Float32(dX_46_u * dX_46_u) * t_3)), fma(Float32(t_5 * dY_46_v), dY_46_v, Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))))));
	else
		tmp = fma(Float32(dY_46_u / sqrt(fmax(fma(Float32(t_5 * dX_46_v), dX_46_v, Float32(t_3 * Float32(dX_46_u * dX_46_u))), fma(Float32(t_3 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_5))))), floor(w), Float32(0.0));
	end
	return tmp
end

Derivation

1. Initial program 75.9%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

5. Applied rewrites 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v}, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   2. lift-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \cdot dX.v, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   3. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   4. *-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   5. associate-*l* N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

   7. lift-*.f32 75.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot \left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]

7. Applied rewrites 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{\left(-dX.u\right) \cdot \left\lfloor w\right\rfloor }{-\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor }, \left\lfloor h\right\rfloor , \left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right), \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}}, \left\lfloor w\right\rfloor , 0\right)\\ \end{array} \]
8. Add Preprocessing

Alternative 7: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := \mathsf{fma}\left(t\_0 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_1\right)\\ t_3 := \mathsf{fma}\left(t\_1 \cdot dX.v, dX.v, t\_0 \cdot \left(dX.u \cdot dX.u\right)\right)\\ t_4 := \frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(t\_3, t\_2\right)}}\\ \mathbf{if}\;t\_3 \geq t\_2:\\ \;\;\;\;t\_4 \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot dY.u\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) (floor w)))
        (t_1 (* (floor h) (floor h)))
        (t_2 (fma (* t_0 dY.u) dY.u (* (* dY.v dY.v) t_1)))
        (t_3 (fma (* t_1 dX.v) dX.v (* t_0 (* dX.u dX.u))))
        (t_4 (/ (floor w) (sqrt (fmax t_3 t_2)))))
   (if (>= t_3 t_2) (* t_4 dX.u) (* t_4 dY.u))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(w) * floorf(w);
	float t_1 = floorf(h) * floorf(h);
	float t_2 = fmaf((t_0 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_1));
	float t_3 = fmaf((t_1 * dX_46_v), dX_46_v, (t_0 * (dX_46_u * dX_46_u)));
	float t_4 = floorf(w) / sqrtf(fmaxf(t_3, t_2));
	float tmp;
	if (t_3 >= t_2) {
		tmp = t_4 * dX_46_u;
	} else {
		tmp = t_4 * dY_46_u;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * floor(w))
	t_1 = Float32(floor(h) * floor(h))
	t_2 = fma(Float32(t_0 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_1))
	t_3 = fma(Float32(t_1 * dX_46_v), dX_46_v, Float32(t_0 * Float32(dX_46_u * dX_46_u)))
	t_4 = Float32(floor(w) / sqrt(fmax(t_3, t_2)))
	tmp = Float32(0.0)
	if (t_3 >= t_2)
		tmp = Float32(t_4 * dX_46_u);
	else
		tmp = Float32(t_4 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 75.9%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 75.8%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right) \geq \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right)\right), \mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)\right)\right)}} \cdot dY.u\\ } \end{array}} \]
4. Add Preprocessing

Alternative 8: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := \left(t\_3 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_5 := \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \\ t_6 := t\_1 \cdot t\_1 + t\_0 \cdot t\_0\\ t_7 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_8 := t\_2 \cdot t\_2 + t\_7 \cdot t\_7\\ t_9 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_10 := \mathsf{fma}\left(t\_9 \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\\ t_11 := \sqrt{\mathsf{max}\left(t\_5, t\_10\right)}\\ t_12 := \frac{1}{\sqrt{\mathsf{max}\left(t\_6, t\_8\right)}}\\ t_13 := \begin{array}{l} \mathbf{if}\;t\_6 \geq t\_8:\\ \;\;\;\;t\_12 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_12 \cdot t\_2\\ \end{array}\\ t_14 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_15 := \mathsf{fma}\left(dY.v \cdot dY.v, t\_9, \left(t\_14 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ t_16 := \sqrt{\mathsf{max}\left(t\_4, t\_15\right)}\\ t_17 := \begin{array}{l} \mathbf{if}\;t\_4 \geq t\_15:\\ \;\;\;\;\frac{t\_3}{t\_16}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_14}{t\_16}\\ \end{array}\\ \mathbf{if}\;t\_13 \leq -0.5:\\ \;\;\;\;t\_17\\ \mathbf{elif}\;t\_13 \leq 0.6000000238418579:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_10:\\ \;\;\;\;\frac{t\_1}{t\_11}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_11}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_17\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) dY.u))
        (t_3 (* dX.u (floor w)))
        (t_4 (* (* t_3 dX.u) (floor w)))
        (t_5 (* (* (* dX.v (floor h)) dX.v) (floor h)))
        (t_6 (+ (* t_1 t_1) (* t_0 t_0)))
        (t_7 (* (floor h) dY.v))
        (t_8 (+ (* t_2 t_2) (* t_7 t_7)))
        (t_9 (* (floor h) (floor h)))
        (t_10
         (fma (* t_9 dY.v) dY.v (* (* (* dY.u dY.u) (floor w)) (floor w))))
        (t_11 (sqrt (fmax t_5 t_10)))
        (t_12 (/ 1.0 (sqrt (fmax t_6 t_8))))
        (t_13 (if (>= t_6 t_8) (* t_12 t_1) (* t_12 t_2)))
        (t_14 (* dY.u (floor w)))
        (t_15 (fma (* dY.v dY.v) t_9 (* (* t_14 dY.u) (floor w))))
        (t_16 (sqrt (fmax t_4 t_15)))
        (t_17 (if (>= t_4 t_15) (/ t_3 t_16) (/ t_14 t_16))))
   (if (<= t_13 -0.5)
     t_17
     (if (<= t_13 0.6000000238418579)
       (if (>= t_5 t_10) (/ t_1 t_11) (/ t_2 t_11))
       t_17))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = (t_3 * dX_46_u) * floorf(w);
	float t_5 = ((dX_46_v * floorf(h)) * dX_46_v) * floorf(h);
	float t_6 = (t_1 * t_1) + (t_0 * t_0);
	float t_7 = floorf(h) * dY_46_v;
	float t_8 = (t_2 * t_2) + (t_7 * t_7);
	float t_9 = floorf(h) * floorf(h);
	float t_10 = fmaf((t_9 * dY_46_v), dY_46_v, (((dY_46_u * dY_46_u) * floorf(w)) * floorf(w)));
	float t_11 = sqrtf(fmaxf(t_5, t_10));
	float t_12 = 1.0f / sqrtf(fmaxf(t_6, t_8));
	float tmp;
	if (t_6 >= t_8) {
		tmp = t_12 * t_1;
	} else {
		tmp = t_12 * t_2;
	}
	float t_13 = tmp;
	float t_14 = dY_46_u * floorf(w);
	float t_15 = fmaf((dY_46_v * dY_46_v), t_9, ((t_14 * dY_46_u) * floorf(w)));
	float t_16 = sqrtf(fmaxf(t_4, t_15));
	float tmp_1;
	if (t_4 >= t_15) {
		tmp_1 = t_3 / t_16;
	} else {
		tmp_1 = t_14 / t_16;
	}
	float t_17 = tmp_1;
	float tmp_2;
	if (t_13 <= -0.5f) {
		tmp_2 = t_17;
	} else if (t_13 <= 0.6000000238418579f) {
		float tmp_3;
		if (t_5 >= t_10) {
			tmp_3 = t_1 / t_11;
		} else {
			tmp_3 = t_2 / t_11;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_17;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(Float32(t_3 * dX_46_u) * floor(w))
	t_5 = Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))
	t_6 = Float32(Float32(t_1 * t_1) + Float32(t_0 * t_0))
	t_7 = Float32(floor(h) * dY_46_v)
	t_8 = Float32(Float32(t_2 * t_2) + Float32(t_7 * t_7))
	t_9 = Float32(floor(h) * floor(h))
	t_10 = fma(Float32(t_9 * dY_46_v), dY_46_v, Float32(Float32(Float32(dY_46_u * dY_46_u) * floor(w)) * floor(w)))
	t_11 = sqrt(fmax(t_5, t_10))
	t_12 = Float32(Float32(1.0) / sqrt(fmax(t_6, t_8)))
	tmp = Float32(0.0)
	if (t_6 >= t_8)
		tmp = Float32(t_12 * t_1);
	else
		tmp = Float32(t_12 * t_2);
	end
	t_13 = tmp
	t_14 = Float32(dY_46_u * floor(w))
	t_15 = fma(Float32(dY_46_v * dY_46_v), t_9, Float32(Float32(t_14 * dY_46_u) * floor(w)))
	t_16 = sqrt(fmax(t_4, t_15))
	tmp_1 = Float32(0.0)
	if (t_4 >= t_15)
		tmp_1 = Float32(t_3 / t_16);
	else
		tmp_1 = Float32(t_14 / t_16);
	end
	t_17 = tmp_1
	tmp_2 = Float32(0.0)
	if (t_13 <= Float32(-0.5))
		tmp_2 = t_17;
	elseif (t_13 <= Float32(0.6000000238418579))
		tmp_3 = Float32(0.0)
		if (t_5 >= t_10)
			tmp_3 = Float32(t_1 / t_11);
		else
			tmp_3 = Float32(t_2 / t_11);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_17;
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.5 or 0.600000024 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 91.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 91.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 93.3%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 93.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 97.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 97.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   if -0.5 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.600000024

   1. Initial program 62.0%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 62.0

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 62.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 61.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 61.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 62.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 62.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   12. Applied rewrites 62.4%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, dY.v, \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ } \end{array}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_5 := \left(t\_4 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_6 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_7 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_8 := t\_3 \cdot t\_3 + t\_7 \cdot t\_7\\ t_9 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_10 := t\_9 \cdot \left(dX.v \cdot dX.v\right)\\ t_11 := \frac{1}{\sqrt{\mathsf{max}\left(t\_6, t\_8\right)}}\\ t_12 := \begin{array}{l} \mathbf{if}\;t\_6 \geq t\_8:\\ \;\;\;\;t\_11 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_11 \cdot t\_3\\ \end{array}\\ t_13 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_14 := \mathsf{fma}\left(dY.v \cdot dY.v, t\_9, \left(t\_13 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ t_15 := \sqrt{\mathsf{max}\left(t\_5, t\_14\right)}\\ t_16 := \begin{array}{l} \mathbf{if}\;t\_5 \geq t\_14:\\ \;\;\;\;\frac{t\_4}{t\_15}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_13}{t\_15}\\ \end{array}\\ t_17 := \frac{1}{\sqrt{\mathsf{max}\left(t\_10, t\_1\right)}}\\ \mathbf{if}\;t\_12 \leq -0.5:\\ \;\;\;\;t\_16\\ \mathbf{elif}\;t\_12 \leq 0.004999999888241291:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_10 \geq t\_1:\\ \;\;\;\;t\_17 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_17 \cdot t\_3\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_16\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (* (* dY.v (floor h)) dY.v) (floor h)))
        (t_2 (* (floor w) dX.u))
        (t_3 (* (floor w) dY.u))
        (t_4 (* dX.u (floor w)))
        (t_5 (* (* t_4 dX.u) (floor w)))
        (t_6 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_7 (* (floor h) dY.v))
        (t_8 (+ (* t_3 t_3) (* t_7 t_7)))
        (t_9 (* (floor h) (floor h)))
        (t_10 (* t_9 (* dX.v dX.v)))
        (t_11 (/ 1.0 (sqrt (fmax t_6 t_8))))
        (t_12 (if (>= t_6 t_8) (* t_11 t_2) (* t_11 t_3)))
        (t_13 (* dY.u (floor w)))
        (t_14 (fma (* dY.v dY.v) t_9 (* (* t_13 dY.u) (floor w))))
        (t_15 (sqrt (fmax t_5 t_14)))
        (t_16 (if (>= t_5 t_14) (/ t_4 t_15) (/ t_13 t_15)))
        (t_17 (/ 1.0 (sqrt (fmax t_10 t_1)))))
   (if (<= t_12 -0.5)
     t_16
     (if (<= t_12 0.004999999888241291)
       (if (>= t_10 t_1) (* t_17 t_2) (* t_17 t_3))
       t_16))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = ((dY_46_v * floorf(h)) * dY_46_v) * floorf(h);
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = dX_46_u * floorf(w);
	float t_5 = (t_4 * dX_46_u) * floorf(w);
	float t_6 = (t_2 * t_2) + (t_0 * t_0);
	float t_7 = floorf(h) * dY_46_v;
	float t_8 = (t_3 * t_3) + (t_7 * t_7);
	float t_9 = floorf(h) * floorf(h);
	float t_10 = t_9 * (dX_46_v * dX_46_v);
	float t_11 = 1.0f / sqrtf(fmaxf(t_6, t_8));
	float tmp;
	if (t_6 >= t_8) {
		tmp = t_11 * t_2;
	} else {
		tmp = t_11 * t_3;
	}
	float t_12 = tmp;
	float t_13 = dY_46_u * floorf(w);
	float t_14 = fmaf((dY_46_v * dY_46_v), t_9, ((t_13 * dY_46_u) * floorf(w)));
	float t_15 = sqrtf(fmaxf(t_5, t_14));
	float tmp_1;
	if (t_5 >= t_14) {
		tmp_1 = t_4 / t_15;
	} else {
		tmp_1 = t_13 / t_15;
	}
	float t_16 = tmp_1;
	float t_17 = 1.0f / sqrtf(fmaxf(t_10, t_1));
	float tmp_2;
	if (t_12 <= -0.5f) {
		tmp_2 = t_16;
	} else if (t_12 <= 0.004999999888241291f) {
		float tmp_3;
		if (t_10 >= t_1) {
			tmp_3 = t_17 * t_2;
		} else {
			tmp_3 = t_17 * t_3;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_16;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = Float32(Float32(t_4 * dX_46_u) * floor(w))
	t_6 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_7 = Float32(floor(h) * dY_46_v)
	t_8 = Float32(Float32(t_3 * t_3) + Float32(t_7 * t_7))
	t_9 = Float32(floor(h) * floor(h))
	t_10 = Float32(t_9 * Float32(dX_46_v * dX_46_v))
	t_11 = Float32(Float32(1.0) / sqrt(fmax(t_6, t_8)))
	tmp = Float32(0.0)
	if (t_6 >= t_8)
		tmp = Float32(t_11 * t_2);
	else
		tmp = Float32(t_11 * t_3);
	end
	t_12 = tmp
	t_13 = Float32(dY_46_u * floor(w))
	t_14 = fma(Float32(dY_46_v * dY_46_v), t_9, Float32(Float32(t_13 * dY_46_u) * floor(w)))
	t_15 = sqrt(fmax(t_5, t_14))
	tmp_1 = Float32(0.0)
	if (t_5 >= t_14)
		tmp_1 = Float32(t_4 / t_15);
	else
		tmp_1 = Float32(t_13 / t_15);
	end
	t_16 = tmp_1
	t_17 = Float32(Float32(1.0) / sqrt(fmax(t_10, t_1)))
	tmp_2 = Float32(0.0)
	if (t_12 <= Float32(-0.5))
		tmp_2 = t_16;
	elseif (t_12 <= Float32(0.004999999888241291))
		tmp_3 = Float32(0.0)
		if (t_10 >= t_1)
			tmp_3 = Float32(t_17 * t_2);
		else
			tmp_3 = Float32(t_17 * t_3);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_16;
	end
	return tmp_2
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.5 or 0.00499999989 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 89.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 89.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 92.1%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 92.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 95.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 96.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}}\\ \end{array} \]

   if -0.5 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.00499999989

   1. Initial program 61.1%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 61.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 61.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.7

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 60.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 61.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 61.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 61.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 61.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   17. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   19. Applied rewrites 60.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := \left(t\_3 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_5 := \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\\ t_6 := \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \\ t_7 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_8 := t\_2 \cdot t\_2 + t\_7 \cdot t\_7\\ t_9 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_10 := t\_9 \cdot t\_9 + t\_0 \cdot t\_0\\ t_11 := \frac{1}{\sqrt{\mathsf{max}\left(t\_10, t\_8\right)}}\\ t_12 := \begin{array}{l} \mathbf{if}\;t\_10 \geq t\_8:\\ \;\;\;\;t\_11 \cdot t\_9\\ \mathbf{else}:\\ \;\;\;\;t\_11 \cdot t\_2\\ \end{array}\\ t_13 := \sqrt{\mathsf{max}\left(t\_4, t\_5\right)}\\ t_14 := \begin{array}{l} \mathbf{if}\;t\_4 \geq t\_5:\\ \;\;\;\;\frac{t\_3}{t\_13}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{t\_13}\\ \end{array}\\ t_15 := \frac{1}{\sqrt{\mathsf{max}\left(t\_1, t\_6\right)}}\\ \mathbf{if}\;t\_12 \leq -0.5:\\ \;\;\;\;t\_14\\ \mathbf{elif}\;t\_12 \leq 0.6000000238418579:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_1 \geq t\_6:\\ \;\;\;\;t\_15 \cdot t\_9\\ \mathbf{else}:\\ \;\;\;\;t\_15 \cdot t\_2\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_14\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (* (floor h) (floor h)) (* dX.v dX.v)))
        (t_2 (* (floor w) dY.u))
        (t_3 (* dX.u (floor w)))
        (t_4 (* (* t_3 dX.u) (floor w)))
        (t_5 (* (* (floor w) (floor w)) (* dY.u dY.u)))
        (t_6 (* (* (* dY.v (floor h)) dY.v) (floor h)))
        (t_7 (* (floor h) dY.v))
        (t_8 (+ (* t_2 t_2) (* t_7 t_7)))
        (t_9 (* (floor w) dX.u))
        (t_10 (+ (* t_9 t_9) (* t_0 t_0)))
        (t_11 (/ 1.0 (sqrt (fmax t_10 t_8))))
        (t_12 (if (>= t_10 t_8) (* t_11 t_9) (* t_11 t_2)))
        (t_13 (sqrt (fmax t_4 t_5)))
        (t_14 (if (>= t_4 t_5) (/ t_3 t_13) (/ (* dY.u (floor w)) t_13)))
        (t_15 (/ 1.0 (sqrt (fmax t_1 t_6)))))
   (if (<= t_12 -0.5)
     t_14
     (if (<= t_12 0.6000000238418579)
       (if (>= t_1 t_6) (* t_15 t_9) (* t_15 t_2))
       t_14))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = (floorf(h) * floorf(h)) * (dX_46_v * dX_46_v);
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_u * floorf(w);
	float t_4 = (t_3 * dX_46_u) * floorf(w);
	float t_5 = (floorf(w) * floorf(w)) * (dY_46_u * dY_46_u);
	float t_6 = ((dY_46_v * floorf(h)) * dY_46_v) * floorf(h);
	float t_7 = floorf(h) * dY_46_v;
	float t_8 = (t_2 * t_2) + (t_7 * t_7);
	float t_9 = floorf(w) * dX_46_u;
	float t_10 = (t_9 * t_9) + (t_0 * t_0);
	float t_11 = 1.0f / sqrtf(fmaxf(t_10, t_8));
	float tmp;
	if (t_10 >= t_8) {
		tmp = t_11 * t_9;
	} else {
		tmp = t_11 * t_2;
	}
	float t_12 = tmp;
	float t_13 = sqrtf(fmaxf(t_4, t_5));
	float tmp_1;
	if (t_4 >= t_5) {
		tmp_1 = t_3 / t_13;
	} else {
		tmp_1 = (dY_46_u * floorf(w)) / t_13;
	}
	float t_14 = tmp_1;
	float t_15 = 1.0f / sqrtf(fmaxf(t_1, t_6));
	float tmp_2;
	if (t_12 <= -0.5f) {
		tmp_2 = t_14;
	} else if (t_12 <= 0.6000000238418579f) {
		float tmp_3;
		if (t_1 >= t_6) {
			tmp_3 = t_15 * t_9;
		} else {
			tmp_3 = t_15 * t_2;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_14;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(Float32(t_3 * dX_46_u) * floor(w))
	t_5 = Float32(Float32(floor(w) * floor(w)) * Float32(dY_46_u * dY_46_u))
	t_6 = Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))
	t_7 = Float32(floor(h) * dY_46_v)
	t_8 = Float32(Float32(t_2 * t_2) + Float32(t_7 * t_7))
	t_9 = Float32(floor(w) * dX_46_u)
	t_10 = Float32(Float32(t_9 * t_9) + Float32(t_0 * t_0))
	t_11 = Float32(Float32(1.0) / sqrt(fmax(t_10, t_8)))
	tmp = Float32(0.0)
	if (t_10 >= t_8)
		tmp = Float32(t_11 * t_9);
	else
		tmp = Float32(t_11 * t_2);
	end
	t_12 = tmp
	t_13 = sqrt(fmax(t_4, t_5))
	tmp_1 = Float32(0.0)
	if (t_4 >= t_5)
		tmp_1 = Float32(t_3 / t_13);
	else
		tmp_1 = Float32(Float32(dY_46_u * floor(w)) / t_13);
	end
	t_14 = tmp_1
	t_15 = Float32(Float32(1.0) / sqrt(fmax(t_1, t_6)))
	tmp_2 = Float32(0.0)
	if (t_12 <= Float32(-0.5))
		tmp_2 = t_14;
	elseif (t_12 <= Float32(0.6000000238418579))
		tmp_3 = Float32(0.0)
		if (t_1 >= t_6)
			tmp_3 = Float32(t_15 * t_9);
		else
			tmp_3 = Float32(t_15 * t_2);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_14;
	end
	return tmp_2
end

MATLAB

function tmp_5 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = (floor(h) * floor(h)) * (dX_46_v * dX_46_v);
	t_2 = floor(w) * dY_46_u;
	t_3 = dX_46_u * floor(w);
	t_4 = (t_3 * dX_46_u) * floor(w);
	t_5 = (floor(w) * floor(w)) * (dY_46_u * dY_46_u);
	t_6 = ((dY_46_v * floor(h)) * dY_46_v) * floor(h);
	t_7 = floor(h) * dY_46_v;
	t_8 = (t_2 * t_2) + (t_7 * t_7);
	t_9 = floor(w) * dX_46_u;
	t_10 = (t_9 * t_9) + (t_0 * t_0);
	t_11 = single(1.0) / sqrt(max(t_10, t_8));
	tmp = single(0.0);
	if (t_10 >= t_8)
		tmp = t_11 * t_9;
	else
		tmp = t_11 * t_2;
	end
	t_12 = tmp;
	t_13 = sqrt(max(t_4, t_5));
	tmp_2 = single(0.0);
	if (t_4 >= t_5)
		tmp_2 = t_3 / t_13;
	else
		tmp_2 = (dY_46_u * floor(w)) / t_13;
	end
	t_14 = tmp_2;
	t_15 = single(1.0) / sqrt(max(t_1, t_6));
	tmp_3 = single(0.0);
	if (t_12 <= single(-0.5))
		tmp_3 = t_14;
	elseif (t_12 <= single(0.6000000238418579))
		tmp_4 = single(0.0);
		if (t_1 >= t_6)
			tmp_4 = t_15 * t_9;
		else
			tmp_4 = t_15 * t_2;
		end
		tmp_3 = tmp_4;
	else
		tmp_3 = t_14;
	end
	tmp_5 = tmp_3;
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.5 or 0.600000024 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 91.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 91.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 93.3%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   20. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 93.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 93.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   23. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 97.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 97.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor \color{blue}{w}\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   26. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 94.9

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   27. Applied rewrites 94.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor \color{blue}{w}\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   if -0.5 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.600000024

   1. Initial program 62.0%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 62.0

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 62.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 61.1

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 61.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 62.2

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 62.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 61.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 61.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{\left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   17. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   19. Applied rewrites 60.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 58.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_5 := \left(t\_4 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_6 := \sqrt{\mathsf{max}\left(t\_5, t\_2\right)}\\ t_7 := \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\\ t_8 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_9 := t\_3 \cdot t\_3 + t\_8 \cdot t\_8\\ t_10 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_11 := t\_10 \cdot t\_10 + t\_1 \cdot t\_1\\ t_12 := \frac{1}{\sqrt{\mathsf{max}\left(t\_11, t\_9\right)}}\\ t_13 := \begin{array}{l} \mathbf{if}\;t\_11 \geq t\_9:\\ \;\;\;\;t\_12 \cdot t\_10\\ \mathbf{else}:\\ \;\;\;\;t\_12 \cdot t\_3\\ \end{array}\\ t_14 := \sqrt{\mathsf{max}\left(t\_5, t\_7\right)}\\ t_15 := \begin{array}{l} \mathbf{if}\;t\_5 \geq t\_7:\\ \;\;\;\;\frac{t\_4}{t\_14}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_14}\\ \end{array}\\ \mathbf{if}\;t\_13 \leq -0.9959999918937683:\\ \;\;\;\;t\_15\\ \mathbf{elif}\;t\_13 \leq 0.8999999761581421:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_2:\\ \;\;\;\;\frac{t\_4}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_6}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t\_15\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.u (floor w)))
        (t_1 (* (floor h) dX.v))
        (t_2 (* (floor h) (* (floor h) (* dY.v dY.v))))
        (t_3 (* (floor w) dY.u))
        (t_4 (* dX.u (floor w)))
        (t_5 (* (* t_4 dX.u) (floor w)))
        (t_6 (sqrt (fmax t_5 t_2)))
        (t_7 (* (* (floor w) (floor w)) (* dY.u dY.u)))
        (t_8 (* (floor h) dY.v))
        (t_9 (+ (* t_3 t_3) (* t_8 t_8)))
        (t_10 (* (floor w) dX.u))
        (t_11 (+ (* t_10 t_10) (* t_1 t_1)))
        (t_12 (/ 1.0 (sqrt (fmax t_11 t_9))))
        (t_13 (if (>= t_11 t_9) (* t_12 t_10) (* t_12 t_3)))
        (t_14 (sqrt (fmax t_5 t_7)))
        (t_15 (if (>= t_5 t_7) (/ t_4 t_14) (/ t_0 t_14))))
   (if (<= t_13 -0.9959999918937683)
     t_15
     (if (<= t_13 0.8999999761581421)
       (if (>= t_5 t_2) (/ t_4 t_6) (/ t_0 t_6))
       t_15))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dY_46_u * floorf(w);
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = floorf(h) * (floorf(h) * (dY_46_v * dY_46_v));
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = dX_46_u * floorf(w);
	float t_5 = (t_4 * dX_46_u) * floorf(w);
	float t_6 = sqrtf(fmaxf(t_5, t_2));
	float t_7 = (floorf(w) * floorf(w)) * (dY_46_u * dY_46_u);
	float t_8 = floorf(h) * dY_46_v;
	float t_9 = (t_3 * t_3) + (t_8 * t_8);
	float t_10 = floorf(w) * dX_46_u;
	float t_11 = (t_10 * t_10) + (t_1 * t_1);
	float t_12 = 1.0f / sqrtf(fmaxf(t_11, t_9));
	float tmp;
	if (t_11 >= t_9) {
		tmp = t_12 * t_10;
	} else {
		tmp = t_12 * t_3;
	}
	float t_13 = tmp;
	float t_14 = sqrtf(fmaxf(t_5, t_7));
	float tmp_1;
	if (t_5 >= t_7) {
		tmp_1 = t_4 / t_14;
	} else {
		tmp_1 = t_0 / t_14;
	}
	float t_15 = tmp_1;
	float tmp_2;
	if (t_13 <= -0.9959999918937683f) {
		tmp_2 = t_15;
	} else if (t_13 <= 0.8999999761581421f) {
		float tmp_3;
		if (t_5 >= t_2) {
			tmp_3 = t_4 / t_6;
		} else {
			tmp_3 = t_0 / t_6;
		}
		tmp_2 = tmp_3;
	} else {
		tmp_2 = t_15;
	}
	return tmp_2;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_u * floor(w))
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = Float32(Float32(t_4 * dX_46_u) * floor(w))
	t_6 = sqrt(fmax(t_5, t_2))
	t_7 = Float32(Float32(floor(w) * floor(w)) * Float32(dY_46_u * dY_46_u))
	t_8 = Float32(floor(h) * dY_46_v)
	t_9 = Float32(Float32(t_3 * t_3) + Float32(t_8 * t_8))
	t_10 = Float32(floor(w) * dX_46_u)
	t_11 = Float32(Float32(t_10 * t_10) + Float32(t_1 * t_1))
	t_12 = Float32(Float32(1.0) / sqrt(fmax(t_11, t_9)))
	tmp = Float32(0.0)
	if (t_11 >= t_9)
		tmp = Float32(t_12 * t_10);
	else
		tmp = Float32(t_12 * t_3);
	end
	t_13 = tmp
	t_14 = sqrt(fmax(t_5, t_7))
	tmp_1 = Float32(0.0)
	if (t_5 >= t_7)
		tmp_1 = Float32(t_4 / t_14);
	else
		tmp_1 = Float32(t_0 / t_14);
	end
	t_15 = tmp_1
	tmp_2 = Float32(0.0)
	if (t_13 <= Float32(-0.9959999918937683))
		tmp_2 = t_15;
	elseif (t_13 <= Float32(0.8999999761581421))
		tmp_3 = Float32(0.0)
		if (t_5 >= t_2)
			tmp_3 = Float32(t_4 / t_6);
		else
			tmp_3 = Float32(t_0 / t_6);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = t_15;
	end
	return tmp_2
end

MATLAB

function tmp_5 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = dY_46_u * floor(w);
	t_1 = floor(h) * dX_46_v;
	t_2 = floor(h) * (floor(h) * (dY_46_v * dY_46_v));
	t_3 = floor(w) * dY_46_u;
	t_4 = dX_46_u * floor(w);
	t_5 = (t_4 * dX_46_u) * floor(w);
	t_6 = sqrt(max(t_5, t_2));
	t_7 = (floor(w) * floor(w)) * (dY_46_u * dY_46_u);
	t_8 = floor(h) * dY_46_v;
	t_9 = (t_3 * t_3) + (t_8 * t_8);
	t_10 = floor(w) * dX_46_u;
	t_11 = (t_10 * t_10) + (t_1 * t_1);
	t_12 = single(1.0) / sqrt(max(t_11, t_9));
	tmp = single(0.0);
	if (t_11 >= t_9)
		tmp = t_12 * t_10;
	else
		tmp = t_12 * t_3;
	end
	t_13 = tmp;
	t_14 = sqrt(max(t_5, t_7));
	tmp_2 = single(0.0);
	if (t_5 >= t_7)
		tmp_2 = t_4 / t_14;
	else
		tmp_2 = t_0 / t_14;
	end
	t_15 = tmp_2;
	tmp_3 = single(0.0);
	if (t_13 <= single(-0.9959999918937683))
		tmp_3 = t_15;
	elseif (t_13 <= single(0.8999999761581421))
		tmp_4 = single(0.0);
		if (t_5 >= t_2)
			tmp_4 = t_4 / t_6;
		else
			tmp_4 = t_0 / t_6;
		end
		tmp_3 = tmp_4;
	else
		tmp_3 = t_15;
	end
	tmp_5 = tmp_3;
end

Derivation

1. Split input into 2 regimes

2. if (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < -0.995999992 or 0.899999976 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u)))

   1. Initial program 99.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 96.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 93.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 93.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 91.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 91.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 94.2%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   20. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 94.2

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 94.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   23. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 98.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \color{blue}{\left\lfloor w\right\rfloor }\right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 98.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor \color{blue}{w}\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]
   26. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       2. pow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

       3. lift-*.f32 96.9

          \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   27. Applied rewrites 96.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor \color{blue}{w}\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   if -0.995999992 < (if (>=.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dX.u)) (*.f32 (/.f32 #s(literal 1 binary32) (sqrt.f32 (fmax.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v)))))) (*.f32 (floor.f32 w) dY.u))) < 0.899999976

   1. Initial program 62.7%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 61.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 61.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 52.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 36.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 28.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 27.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 25.4%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 25.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 27.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 37.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := \left(t\_2 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_4 := \sqrt{\mathsf{max}\left(t\_3, t\_1\right)}\\ t_5 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_6 := \left(t\_5 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ t_7 := \sqrt{\mathsf{max}\left(t\_0, t\_6\right)}\\ \mathbf{if}\;dX.v \leq 1:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq t\_1:\\ \;\;\;\;\frac{t\_2}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{t\_4}\\ \end{array}\\ \mathbf{elif}\;t\_0 \geq t\_6:\\ \;\;\;\;\frac{t\_2}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{t\_7}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* (* dX.v (floor h)) dX.v) (floor h)))
        (t_1 (* (floor h) (* (floor h) (* dY.v dY.v))))
        (t_2 (* dX.u (floor w)))
        (t_3 (* (* t_2 dX.u) (floor w)))
        (t_4 (sqrt (fmax t_3 t_1)))
        (t_5 (* dY.u (floor w)))
        (t_6 (* (* t_5 dY.u) (floor w)))
        (t_7 (sqrt (fmax t_0 t_6))))
   (if (<= dX.v 1.0)
     (if (>= t_3 t_1) (/ t_2 t_4) (/ t_5 t_4))
     (if (>= t_0 t_6) (/ t_2 t_7) (/ t_5 t_7)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = ((dX_46_v * floorf(h)) * dX_46_v) * floorf(h);
	float t_1 = floorf(h) * (floorf(h) * (dY_46_v * dY_46_v));
	float t_2 = dX_46_u * floorf(w);
	float t_3 = (t_2 * dX_46_u) * floorf(w);
	float t_4 = sqrtf(fmaxf(t_3, t_1));
	float t_5 = dY_46_u * floorf(w);
	float t_6 = (t_5 * dY_46_u) * floorf(w);
	float t_7 = sqrtf(fmaxf(t_0, t_6));
	float tmp_1;
	if (dX_46_v <= 1.0f) {
		float tmp_2;
		if (t_3 >= t_1) {
			tmp_2 = t_2 / t_4;
		} else {
			tmp_2 = t_5 / t_4;
		}
		tmp_1 = tmp_2;
	} else if (t_0 >= t_6) {
		tmp_1 = t_2 / t_7;
	} else {
		tmp_1 = t_5 / t_7;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))
	t_1 = Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(Float32(t_2 * dX_46_u) * floor(w))
	t_4 = sqrt(fmax(t_3, t_1))
	t_5 = Float32(dY_46_u * floor(w))
	t_6 = Float32(Float32(t_5 * dY_46_u) * floor(w))
	t_7 = sqrt(fmax(t_0, t_6))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(1.0))
		tmp_2 = Float32(0.0)
		if (t_3 >= t_1)
			tmp_2 = Float32(t_2 / t_4);
		else
			tmp_2 = Float32(t_5 / t_4);
		end
		tmp_1 = tmp_2;
	elseif (t_0 >= t_6)
		tmp_1 = Float32(t_2 / t_7);
	else
		tmp_1 = Float32(t_5 / t_7);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = ((dX_46_v * floor(h)) * dX_46_v) * floor(h);
	t_1 = floor(h) * (floor(h) * (dY_46_v * dY_46_v));
	t_2 = dX_46_u * floor(w);
	t_3 = (t_2 * dX_46_u) * floor(w);
	t_4 = sqrt(max(t_3, t_1));
	t_5 = dY_46_u * floor(w);
	t_6 = (t_5 * dY_46_u) * floor(w);
	t_7 = sqrt(max(t_0, t_6));
	tmp_2 = single(0.0);
	if (dX_46_v <= single(1.0))
		tmp_3 = single(0.0);
		if (t_3 >= t_1)
			tmp_3 = t_2 / t_4;
		else
			tmp_3 = t_5 / t_4;
		end
		tmp_2 = tmp_3;
	elseif (t_0 >= t_6)
		tmp_2 = t_2 / t_7;
	else
		tmp_2 = t_5 / t_7;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1

   1. Initial program 77.8%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 68.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 61.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 56.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 56.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 54.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 52.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \end{array} \]

   if 1 < dX.v

   1. Initial program 70.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 66.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 66.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 59.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 59.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 60.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 53.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 53.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   17. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   19. Applied rewrites 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   21. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ } \end{array}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := \left(t\_2 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_4 := \sqrt{\mathsf{max}\left(t\_3, t\_1\right)}\\ t_5 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_6 := \left(t\_5 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \mathbf{if}\;dX.v \leq 1:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq t\_1:\\ \;\;\;\;\frac{t\_2}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{t\_4}\\ \end{array}\\ \mathbf{elif}\;t\_0 \geq t\_6:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{max}\left(t\_0, t\_6\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right), t\_6\right)}} \cdot dY.u\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* (* dX.v (floor h)) dX.v) (floor h)))
        (t_1 (* (floor h) (* (floor h) (* dY.v dY.v))))
        (t_2 (* dX.u (floor w)))
        (t_3 (* (* t_2 dX.u) (floor w)))
        (t_4 (sqrt (fmax t_3 t_1)))
        (t_5 (* dY.u (floor w)))
        (t_6 (* (* t_5 dY.u) (floor w))))
   (if (<= dX.v 1.0)
     (if (>= t_3 t_1) (/ t_2 t_4) (/ t_5 t_4))
     (if (>= t_0 t_6)
       (/ t_2 (sqrt (fmax t_0 t_6)))
       (*
        (/
         (floor w)
         (sqrt (fmax (* (floor h) (* (floor h) (* dX.v dX.v))) t_6)))
        dY.u)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = ((dX_46_v * floorf(h)) * dX_46_v) * floorf(h);
	float t_1 = floorf(h) * (floorf(h) * (dY_46_v * dY_46_v));
	float t_2 = dX_46_u * floorf(w);
	float t_3 = (t_2 * dX_46_u) * floorf(w);
	float t_4 = sqrtf(fmaxf(t_3, t_1));
	float t_5 = dY_46_u * floorf(w);
	float t_6 = (t_5 * dY_46_u) * floorf(w);
	float tmp_1;
	if (dX_46_v <= 1.0f) {
		float tmp_2;
		if (t_3 >= t_1) {
			tmp_2 = t_2 / t_4;
		} else {
			tmp_2 = t_5 / t_4;
		}
		tmp_1 = tmp_2;
	} else if (t_0 >= t_6) {
		tmp_1 = t_2 / sqrtf(fmaxf(t_0, t_6));
	} else {
		tmp_1 = (floorf(w) / sqrtf(fmaxf((floorf(h) * (floorf(h) * (dX_46_v * dX_46_v))), t_6))) * dY_46_u;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))
	t_1 = Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(Float32(t_2 * dX_46_u) * floor(w))
	t_4 = sqrt(fmax(t_3, t_1))
	t_5 = Float32(dY_46_u * floor(w))
	t_6 = Float32(Float32(t_5 * dY_46_u) * floor(w))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(1.0))
		tmp_2 = Float32(0.0)
		if (t_3 >= t_1)
			tmp_2 = Float32(t_2 / t_4);
		else
			tmp_2 = Float32(t_5 / t_4);
		end
		tmp_1 = tmp_2;
	elseif (t_0 >= t_6)
		tmp_1 = Float32(t_2 / sqrt(fmax(t_0, t_6)));
	else
		tmp_1 = Float32(Float32(floor(w) / sqrt(fmax(Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v))), t_6))) * dY_46_u);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = ((dX_46_v * floor(h)) * dX_46_v) * floor(h);
	t_1 = floor(h) * (floor(h) * (dY_46_v * dY_46_v));
	t_2 = dX_46_u * floor(w);
	t_3 = (t_2 * dX_46_u) * floor(w);
	t_4 = sqrt(max(t_3, t_1));
	t_5 = dY_46_u * floor(w);
	t_6 = (t_5 * dY_46_u) * floor(w);
	tmp_2 = single(0.0);
	if (dX_46_v <= single(1.0))
		tmp_3 = single(0.0);
		if (t_3 >= t_1)
			tmp_3 = t_2 / t_4;
		else
			tmp_3 = t_5 / t_4;
		end
		tmp_2 = tmp_3;
	elseif (t_0 >= t_6)
		tmp_2 = t_2 / sqrt(max(t_0, t_6));
	else
		tmp_2 = (floor(w) / sqrt(max((floor(h) * (floor(h) * (dX_46_v * dX_46_v))), t_6))) * dY_46_u;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1

   1. Initial program 77.8%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 68.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 61.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 56.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 56.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 54.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 52.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \end{array} \]

   if 1 < dX.v

   1. Initial program 70.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 66.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 66.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 59.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 59.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 60.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 53.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 53.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   17. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   19. Applied rewrites 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   21. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ } \end{array}} \]

   23. Applied rewrites 51.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dY.u\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := \left(t\_2 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_4 := \sqrt{\mathsf{max}\left(t\_3, t\_1\right)}\\ t_5 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_6 := \left(t\_5 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \mathbf{if}\;dX.v \leq 1:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq t\_1:\\ \;\;\;\;\frac{t\_2}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{t\_4}\\ \end{array}\\ \mathbf{elif}\;t\_0 \geq t\_6:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{max}\left(\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right), t\_6\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{\sqrt{\mathsf{max}\left(t\_0, t\_6\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* (* dX.v (floor h)) dX.v) (floor h)))
        (t_1 (* (floor h) (* (floor h) (* dY.v dY.v))))
        (t_2 (* dX.u (floor w)))
        (t_3 (* (* t_2 dX.u) (floor w)))
        (t_4 (sqrt (fmax t_3 t_1)))
        (t_5 (* dY.u (floor w)))
        (t_6 (* (* t_5 dY.u) (floor w))))
   (if (<= dX.v 1.0)
     (if (>= t_3 t_1) (/ t_2 t_4) (/ t_5 t_4))
     (if (>= t_0 t_6)
       (/ t_2 (sqrt (fmax (* (floor h) (* (floor h) (* dX.v dX.v))) t_6)))
       (/ t_5 (sqrt (fmax t_0 t_6)))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = ((dX_46_v * floorf(h)) * dX_46_v) * floorf(h);
	float t_1 = floorf(h) * (floorf(h) * (dY_46_v * dY_46_v));
	float t_2 = dX_46_u * floorf(w);
	float t_3 = (t_2 * dX_46_u) * floorf(w);
	float t_4 = sqrtf(fmaxf(t_3, t_1));
	float t_5 = dY_46_u * floorf(w);
	float t_6 = (t_5 * dY_46_u) * floorf(w);
	float tmp_1;
	if (dX_46_v <= 1.0f) {
		float tmp_2;
		if (t_3 >= t_1) {
			tmp_2 = t_2 / t_4;
		} else {
			tmp_2 = t_5 / t_4;
		}
		tmp_1 = tmp_2;
	} else if (t_0 >= t_6) {
		tmp_1 = t_2 / sqrtf(fmaxf((floorf(h) * (floorf(h) * (dX_46_v * dX_46_v))), t_6));
	} else {
		tmp_1 = t_5 / sqrtf(fmaxf(t_0, t_6));
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))
	t_1 = Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(Float32(t_2 * dX_46_u) * floor(w))
	t_4 = sqrt(fmax(t_3, t_1))
	t_5 = Float32(dY_46_u * floor(w))
	t_6 = Float32(Float32(t_5 * dY_46_u) * floor(w))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(1.0))
		tmp_2 = Float32(0.0)
		if (t_3 >= t_1)
			tmp_2 = Float32(t_2 / t_4);
		else
			tmp_2 = Float32(t_5 / t_4);
		end
		tmp_1 = tmp_2;
	elseif (t_0 >= t_6)
		tmp_1 = Float32(t_2 / sqrt(fmax(Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v))), t_6)));
	else
		tmp_1 = Float32(t_5 / sqrt(fmax(t_0, t_6)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = ((dX_46_v * floor(h)) * dX_46_v) * floor(h);
	t_1 = floor(h) * (floor(h) * (dY_46_v * dY_46_v));
	t_2 = dX_46_u * floor(w);
	t_3 = (t_2 * dX_46_u) * floor(w);
	t_4 = sqrt(max(t_3, t_1));
	t_5 = dY_46_u * floor(w);
	t_6 = (t_5 * dY_46_u) * floor(w);
	tmp_2 = single(0.0);
	if (dX_46_v <= single(1.0))
		tmp_3 = single(0.0);
		if (t_3 >= t_1)
			tmp_3 = t_2 / t_4;
		else
			tmp_3 = t_5 / t_4;
		end
		tmp_2 = tmp_3;
	elseif (t_0 >= t_6)
		tmp_2 = t_2 / sqrt(max((floor(h) * (floor(h) * (dX_46_v * dX_46_v))), t_6));
	else
		tmp_2 = t_5 / sqrt(max(t_0, t_6));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1

   1. Initial program 77.8%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 68.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 61.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 56.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 56.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 54.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 52.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \end{array} \]

   if 1 < dX.v

   1. Initial program 70.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 66.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 66.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 59.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 59.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 60.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 53.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 53.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   17. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   19. Applied rewrites 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   21. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ } \end{array}} \]

   23. Applied rewrites 51.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\color{blue}{\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_2 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_3 := \left(t\_2 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_4 := \sqrt{\mathsf{max}\left(t\_3, t\_1\right)}\\ t_5 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_6 := \left(t\_5 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \mathbf{if}\;dX.v \leq 1:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq t\_1:\\ \;\;\;\;\frac{t\_2}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{t\_4}\\ \end{array}\\ \mathbf{elif}\;t\_0 \geq t\_6:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right), t\_6\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_5}{\sqrt{\mathsf{max}\left(t\_0, t\_6\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* (* dX.v (floor h)) dX.v) (floor h)))
        (t_1 (* (floor h) (* (floor h) (* dY.v dY.v))))
        (t_2 (* dX.u (floor w)))
        (t_3 (* (* t_2 dX.u) (floor w)))
        (t_4 (sqrt (fmax t_3 t_1)))
        (t_5 (* dY.u (floor w)))
        (t_6 (* (* t_5 dY.u) (floor w))))
   (if (<= dX.v 1.0)
     (if (>= t_3 t_1) (/ t_2 t_4) (/ t_5 t_4))
     (if (>= t_0 t_6)
       (*
        (/
         (floor w)
         (sqrt (fmax (* (floor h) (* (floor h) (* dX.v dX.v))) t_6)))
        dX.u)
       (/ t_5 (sqrt (fmax t_0 t_6)))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = ((dX_46_v * floorf(h)) * dX_46_v) * floorf(h);
	float t_1 = floorf(h) * (floorf(h) * (dY_46_v * dY_46_v));
	float t_2 = dX_46_u * floorf(w);
	float t_3 = (t_2 * dX_46_u) * floorf(w);
	float t_4 = sqrtf(fmaxf(t_3, t_1));
	float t_5 = dY_46_u * floorf(w);
	float t_6 = (t_5 * dY_46_u) * floorf(w);
	float tmp_1;
	if (dX_46_v <= 1.0f) {
		float tmp_2;
		if (t_3 >= t_1) {
			tmp_2 = t_2 / t_4;
		} else {
			tmp_2 = t_5 / t_4;
		}
		tmp_1 = tmp_2;
	} else if (t_0 >= t_6) {
		tmp_1 = (floorf(w) / sqrtf(fmaxf((floorf(h) * (floorf(h) * (dX_46_v * dX_46_v))), t_6))) * dX_46_u;
	} else {
		tmp_1 = t_5 / sqrtf(fmaxf(t_0, t_6));
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(Float32(Float32(dX_46_v * floor(h)) * dX_46_v) * floor(h))
	t_1 = Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(Float32(t_2 * dX_46_u) * floor(w))
	t_4 = sqrt(fmax(t_3, t_1))
	t_5 = Float32(dY_46_u * floor(w))
	t_6 = Float32(Float32(t_5 * dY_46_u) * floor(w))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(1.0))
		tmp_2 = Float32(0.0)
		if (t_3 >= t_1)
			tmp_2 = Float32(t_2 / t_4);
		else
			tmp_2 = Float32(t_5 / t_4);
		end
		tmp_1 = tmp_2;
	elseif (t_0 >= t_6)
		tmp_1 = Float32(Float32(floor(w) / sqrt(fmax(Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v))), t_6))) * dX_46_u);
	else
		tmp_1 = Float32(t_5 / sqrt(fmax(t_0, t_6)));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = ((dX_46_v * floor(h)) * dX_46_v) * floor(h);
	t_1 = floor(h) * (floor(h) * (dY_46_v * dY_46_v));
	t_2 = dX_46_u * floor(w);
	t_3 = (t_2 * dX_46_u) * floor(w);
	t_4 = sqrt(max(t_3, t_1));
	t_5 = dY_46_u * floor(w);
	t_6 = (t_5 * dY_46_u) * floor(w);
	tmp_2 = single(0.0);
	if (dX_46_v <= single(1.0))
		tmp_3 = single(0.0);
		if (t_3 >= t_1)
			tmp_3 = t_2 / t_4;
		else
			tmp_3 = t_5 / t_4;
		end
		tmp_2 = tmp_3;
	elseif (t_0 >= t_6)
		tmp_2 = (floor(w) / sqrt(max((floor(h) * (floor(h) * (dX_46_v * dX_46_v))), t_6))) * dX_46_u;
	else
		tmp_2 = t_5 / sqrt(max(t_0, t_6));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1

   1. Initial program 77.8%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Applied rewrites 76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 68.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 61.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 56.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dX.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 56.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   18. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

   19. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   21. Applied rewrites 46.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   22. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   24. Applied rewrites 54.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

   25. Taylor expanded in dY.u around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

   27. Applied rewrites 52.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \end{array} \]

   if 1 < dX.v

   1. Initial program 70.4%

      \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 66.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. Applied rewrites 66.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{{dX.v}^{2}}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {\color{blue}{dX.v}}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 59.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \color{blue}{dX.v}\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. Applied rewrites 59.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      2. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      3. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      5. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      6. lift-floor.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot {dX.v}^{2}, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      8. lower-*.f32 60.9

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. Applied rewrites 60.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. Applied rewrites 53.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   16. Applied rewrites 53.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \color{blue}{\left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor }\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   17. Taylor expanded in dY.u around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   19. Applied rewrites 51.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.u \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   21. Applied rewrites 51.6%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ } \end{array}} \]

   23. Applied rewrites 51.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\color{blue}{\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}} \cdot dX.u}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 47.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left(t\_0 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_2 := \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\\ t_3 := \sqrt{\mathsf{max}\left(t\_1, t\_2\right)}\\ \mathbf{if}\;t\_1 \geq t\_2:\\ \;\;\;\;\frac{t\_0}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (* t_0 dX.u) (floor w)))
        (t_2 (* (floor h) (* (floor h) (* dY.v dY.v))))
        (t_3 (sqrt (fmax t_1 t_2))))
   (if (>= t_1 t_2) (/ t_0 t_3) (/ (* dY.u (floor w)) t_3))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = (t_0 * dX_46_u) * floorf(w);
	float t_2 = floorf(h) * (floorf(h) * (dY_46_v * dY_46_v));
	float t_3 = sqrtf(fmaxf(t_1, t_2));
	float tmp;
	if (t_1 >= t_2) {
		tmp = t_0 / t_3;
	} else {
		tmp = (dY_46_u * floorf(w)) / t_3;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(Float32(t_0 * dX_46_u) * floor(w))
	t_2 = Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))
	t_3 = sqrt(fmax(t_1, t_2))
	tmp = Float32(0.0)
	if (t_1 >= t_2)
		tmp = Float32(t_0 / t_3);
	else
		tmp = Float32(Float32(dY_46_u * floor(w)) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = dX_46_u * floor(w);
	t_1 = (t_0 * dX_46_u) * floor(w);
	t_2 = floor(h) * (floor(h) * (dY_46_v * dY_46_v));
	t_3 = sqrt(max(t_1, t_2));
	tmp = single(0.0);
	if (t_1 >= t_2)
		tmp = t_0 / t_3;
	else
		tmp = (dY_46_u * floor(w)) / t_3;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.9%

   \[\begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

3. Applied rewrites 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Applied rewrites 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \color{blue}{\mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Applied rewrites 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Taylor expanded in dX.u around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

10. Applied rewrites 56.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

11. Taylor expanded in dX.u around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

13. Applied rewrites 51.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

14. Taylor expanded in dX.u around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

16. Applied rewrites 50.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(dX.u \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(\left(dY.u \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor h\right\rfloor \right), dY.v \cdot \frac{dY.v}{dY.u \cdot dY.u}, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

18. Applied rewrites 50.1%

    \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ } \end{array}} \]

19. Taylor expanded in dY.u around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

21. Applied rewrites 41.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}:\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

22. Taylor expanded in dY.u around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

24. Applied rewrites 48.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \color{blue}{\left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v, \frac{dY.v}{dY.u \cdot dY.u}, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot dY.u\right)\right)}}\\ \end{array} \]

25. Taylor expanded in dY.u around 0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \end{array} \]

27. Applied rewrites 47.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \geq \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right):\\ \;\;\;\;\frac{dX.u \cdot \left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.u \cdot \color{blue}{\left\lfloor w\right\rfloor }}{\sqrt{\mathsf{max}\left(\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right)\right)\right)}}\\ \end{array} \]
28. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dX.u)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dY.u))))