Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 76.3% → 76.3%

Time: 9.4s

Alternatives: 8

Speedup: 1.0×

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: 76.3% 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: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

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) * floorf(h);
	float t_1 = floorf(w) * floorf(w);
	float t_2 = fmaf((t_1 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_0));
	float t_3 = fmaf((t_1 * dX_46_u), dX_46_u, (t_0 * (dX_46_v * dX_46_v)));
	float t_4 = sqrtf(fmaxf(t_3, t_2));
	float tmp;
	if (t_3 >= t_2) {
		tmp = (floorf(w) * dX_46_u) / t_4;
	} else {
		tmp = (floorf(w) * dY_46_u) / t_4;
	}
	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) * floor(h))
	t_1 = Float32(floor(w) * floor(w))
	t_2 = fma(Float32(t_1 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_0))
	t_3 = fma(Float32(t_1 * dX_46_u), dX_46_u, Float32(t_0 * Float32(dX_46_v * dX_46_v)))
	t_4 = sqrt(fmax(t_3, t_2))
	tmp = Float32(0.0)
	if (t_3 >= t_2)
		tmp = Float32(Float32(floor(w) * dX_46_u) / t_4);
	else
		tmp = Float32(Float32(floor(w) * dY_46_u) / t_4);
	end
	return tmp
end

Derivation

1. Initial program 76.3%

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

3. Applied rewrites 76.3%

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

Alternative 2: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

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) * floorf(h);
	float t_1 = floorf(w) * floorf(w);
	float t_2 = fmaf((t_1 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_0));
	float t_3 = fmaf((t_1 * dX_46_u), dX_46_u, (t_0 * (dX_46_v * dX_46_v)));
	float tmp;
	if (t_3 >= t_2) {
		tmp = (floorf(w) * dX_46_u) / sqrtf(fmaxf(t_3, t_2));
	} else {
		tmp = (dY_46_u / sqrtf(fmaxf(fmaf(((dX_46_u * dX_46_u) * floorf(w)), floorf(w), (((floorf(h) * dX_46_v) * dX_46_v) * floorf(h))), fmaf(((floorf(h) * dY_46_v) * dY_46_v), floorf(h), ((dY_46_u * dY_46_u) * t_1))))) * floorf(w);
	}
	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) * floor(h))
	t_1 = Float32(floor(w) * floor(w))
	t_2 = fma(Float32(t_1 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_0))
	t_3 = fma(Float32(t_1 * dX_46_u), dX_46_u, Float32(t_0 * Float32(dX_46_v * dX_46_v)))
	tmp = Float32(0.0)
	if (t_3 >= t_2)
		tmp = Float32(Float32(floor(w) * dX_46_u) / sqrt(fmax(t_3, t_2)));
	else
		tmp = Float32(Float32(dY_46_u / sqrt(fmax(fma(Float32(Float32(dX_46_u * dX_46_u) * floor(w)), floor(w), Float32(Float32(Float32(floor(h) * dX_46_v) * dX_46_v) * floor(h))), fma(Float32(Float32(floor(h) * dY_46_v) * dY_46_v), floor(h), Float32(Float32(dY_46_u * dY_46_u) * t_1))))) * floor(w));
	end
	return tmp
end

Derivation

1. Initial program 76.3%

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

3. Applied rewrites 76.3%

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

5. Applied rewrites 76.3%

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

Alternative 3: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_1 := \left(dY.v \cdot dY.v\right) \cdot t\_0\\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := \mathsf{fma}\left(t\_2 \cdot dY.u, dY.u, t\_1\right)\\ t_4 := t\_2 \cdot dX.u\\ t_5 := \mathsf{fma}\left(t\_4, dX.u, t\_0 \cdot \left(dX.v \cdot dX.v\right)\right)\\ \mathbf{if}\;t\_5 \geq t\_3:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_4, dX.u, e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right), t\_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) (floor h)))
        (t_1 (* (* dY.v dY.v) t_0))
        (t_2 (* (floor w) (floor w)))
        (t_3 (fma (* t_2 dY.u) dY.u t_1))
        (t_4 (* t_2 dX.u))
        (t_5 (fma t_4 dX.u (* t_0 (* dX.v dX.v)))))
   (if (>= t_5 t_3)
     (/
      (* (floor w) dX.u)
      (sqrt (fmax (fma t_4 dX.u (exp (* (log (* (floor h) dX.v)) 2.0))) t_1)))
     (/ (* (floor w) dY.u) (sqrt (fmax t_5 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 = floorf(h) * floorf(h);
	float t_1 = (dY_46_v * dY_46_v) * t_0;
	float t_2 = floorf(w) * floorf(w);
	float t_3 = fmaf((t_2 * dY_46_u), dY_46_u, t_1);
	float t_4 = t_2 * dX_46_u;
	float t_5 = fmaf(t_4, dX_46_u, (t_0 * (dX_46_v * dX_46_v)));
	float tmp;
	if (t_5 >= t_3) {
		tmp = (floorf(w) * dX_46_u) / sqrtf(fmaxf(fmaf(t_4, dX_46_u, expf((logf((floorf(h) * dX_46_v)) * 2.0f))), t_1));
	} else {
		tmp = (floorf(w) * dY_46_u) / sqrtf(fmaxf(t_5, 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(floor(h) * floor(h))
	t_1 = Float32(Float32(dY_46_v * dY_46_v) * t_0)
	t_2 = Float32(floor(w) * floor(w))
	t_3 = fma(Float32(t_2 * dY_46_u), dY_46_u, t_1)
	t_4 = Float32(t_2 * dX_46_u)
	t_5 = fma(t_4, dX_46_u, Float32(t_0 * Float32(dX_46_v * dX_46_v)))
	tmp = Float32(0.0)
	if (t_5 >= t_3)
		tmp = Float32(Float32(floor(w) * dX_46_u) / sqrt(fmax(fma(t_4, dX_46_u, exp(Float32(log(Float32(floor(h) * dX_46_v)) * Float32(2.0)))), t_1)));
	else
		tmp = Float32(Float32(floor(w) * dY_46_u) / sqrt(fmax(t_5, t_3)));
	end
	return tmp
end

Derivation

1. Initial program 76.3%

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

3. Applied rewrites 76.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-floor.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. unpow-prod-down N/A

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

   9. lift-floor.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. pow-to-exp N/A

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

   12. lower-exp.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-log.f32 59.9

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

5. Applied rewrites 59.9%

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

6. Taylor expanded in dY.u around 0

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

8. Applied rewrites 59.4%

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

Alternative 4: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_4 := \mathsf{fma}\left(t\_0 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_3\right)\\ t_5 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_6 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_7 := \frac{1}{\sqrt{\mathsf{max}\left(t\_2 \cdot t\_2 + t\_6 \cdot t\_6, t\_1 \cdot t\_1 + t\_5 \cdot t\_5\right)}}\\ t_8 := \sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.u, dX.u, t\_3 \cdot \left(dX.v \cdot dX.v\right)\right), t\_4\right)}\\ \mathbf{if}\;dX.v \leq 0.029999999329447746:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot t\_0 \geq \mathsf{fma}\left(t\_5 \cdot dY.v, \left\lfloor h\right\rfloor , \left(dY.u \cdot dY.u\right) \cdot t\_0\right):\\ \;\;\;\;t\_7 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_7 \cdot t\_1\\ \end{array}\\ \mathbf{elif}\;\left(t\_6 \cdot dX.v\right) \cdot \left\lfloor h\right\rfloor \geq t\_4:\\ \;\;\;\;\frac{t\_2}{t\_8}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_8}\\ \end{array} \end{array} \]

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 w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (* (floor h) (floor h)))
        (t_4 (fma (* t_0 dY.u) dY.u (* (* dY.v dY.v) t_3)))
        (t_5 (* (floor h) dY.v))
        (t_6 (* (floor h) dX.v))
        (t_7
         (/
          1.0
          (sqrt
           (fmax (+ (* t_2 t_2) (* t_6 t_6)) (+ (* t_1 t_1) (* t_5 t_5))))))
        (t_8 (sqrt (fmax (fma (* t_0 dX.u) dX.u (* t_3 (* dX.v dX.v))) t_4))))
   (if (<= dX.v 0.029999999329447746)
     (if (>=
          (* (* dX.u dX.u) t_0)
          (fma (* t_5 dY.v) (floor h) (* (* dY.u dY.u) t_0)))
       (* t_7 t_2)
       (* t_7 t_1))
     (if (>= (* (* t_6 dX.v) (floor h)) t_4) (/ t_2 t_8) (/ t_1 t_8)))))

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(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = floorf(h) * floorf(h);
	float t_4 = fmaf((t_0 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_3));
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = floorf(h) * dX_46_v;
	float t_7 = 1.0f / sqrtf(fmaxf(((t_2 * t_2) + (t_6 * t_6)), ((t_1 * t_1) + (t_5 * t_5))));
	float t_8 = sqrtf(fmaxf(fmaf((t_0 * dX_46_u), dX_46_u, (t_3 * (dX_46_v * dX_46_v))), t_4));
	float tmp_1;
	if (dX_46_v <= 0.029999999329447746f) {
		float tmp_2;
		if (((dX_46_u * dX_46_u) * t_0) >= fmaf((t_5 * dY_46_v), floorf(h), ((dY_46_u * dY_46_u) * t_0))) {
			tmp_2 = t_7 * t_2;
		} else {
			tmp_2 = t_7 * t_1;
		}
		tmp_1 = tmp_2;
	} else if (((t_6 * dX_46_v) * floorf(h)) >= t_4) {
		tmp_1 = t_2 / t_8;
	} else {
		tmp_1 = t_1 / t_8;
	}
	return tmp_1;
}

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(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(h) * floor(h))
	t_4 = fma(Float32(t_0 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_3))
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(floor(h) * dX_46_v)
	t_7 = Float32(Float32(1.0) / sqrt(fmax(Float32(Float32(t_2 * t_2) + Float32(t_6 * t_6)), Float32(Float32(t_1 * t_1) + Float32(t_5 * t_5)))))
	t_8 = sqrt(fmax(fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(t_3 * Float32(dX_46_v * dX_46_v))), t_4))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(0.029999999329447746))
		tmp_2 = Float32(0.0)
		if (Float32(Float32(dX_46_u * dX_46_u) * t_0) >= fma(Float32(t_5 * dY_46_v), floor(h), Float32(Float32(dY_46_u * dY_46_u) * t_0)))
			tmp_2 = Float32(t_7 * t_2);
		else
			tmp_2 = Float32(t_7 * t_1);
		end
		tmp_1 = tmp_2;
	elseif (Float32(Float32(t_6 * dX_46_v) * floor(h)) >= t_4)
		tmp_1 = Float32(t_2 / t_8);
	else
		tmp_1 = Float32(t_1 / t_8);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.0299999993

   1. Initial program 77.5%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-*.f32 69.5

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

      6. lift-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot \left(\color{blue}{\left\lfloor w\right\rfloor } \cdot \left\lfloor w\right\rfloor \right) \geq \mathsf{Rewrite=>}\left(lift-+.f32, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

      7. lift-*.f32 N/A

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

   6. Applied rewrites 69.5%

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

   if 0.0299999993 < dX.v

   1. Initial program 72.9%

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in dX.u around 0

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

      1. lift-floor.f32 N/A

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

      2. lift-floor.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 N/A

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

      8. swap-sqr N/A

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

      9. swap-sqr N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   6. Applied rewrites 69.4%

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

Alternative 5: 64.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_1 := \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot dX.v\\ t_2 := t\_1 \cdot \left\lfloor h\right\rfloor \\ t_3 := \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot dY.v\\ t_4 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_5 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_6 := \mathsf{fma}\left(t\_0 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_5\right)\\ t_7 := \sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.u, dX.u, t\_5 \cdot \left(dX.v \cdot dX.v\right)\right), t\_6\right)}\\ \mathbf{if}\;dX.v \leq 0.029999999329447746:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\left(dX.u \cdot dX.u\right) \cdot t\_0 \geq \mathsf{fma}\left(t\_3, \left\lfloor h\right\rfloor , \left(dY.u \cdot dY.u\right) \cdot t\_0\right):\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_1, \left\lfloor h\right\rfloor , \left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \right), \mathsf{fma}\left(dY.v \cdot dY.v, t\_5, \left(t\_4 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}} \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor }{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor w\right\rfloor , t\_2\right), \mathsf{fma}\left(t\_3, \left\lfloor h\right\rfloor , \left(t\_4 \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right)\right)}} \cdot dY.u\\ \end{array}\\ \mathbf{elif}\;t\_2 \geq t\_6:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dX.u}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor w\right\rfloor \cdot dY.u}{t\_7}\\ \end{array} \end{array} \]

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) dX.v) dX.v))
        (t_2 (* t_1 (floor h)))
        (t_3 (* (* (floor h) dY.v) dY.v))
        (t_4 (* dY.u (floor w)))
        (t_5 (* (floor h) (floor h)))
        (t_6 (fma (* t_0 dY.u) dY.u (* (* dY.v dY.v) t_5)))
        (t_7 (sqrt (fmax (fma (* t_0 dX.u) dX.u (* t_5 (* dX.v dX.v))) t_6))))
   (if (<= dX.v 0.029999999329447746)
     (if (>= (* (* dX.u dX.u) t_0) (fma t_3 (floor h) (* (* dY.u dY.u) t_0)))
       (*
        (/
         (floor w)
         (sqrt
          (fmax
           (fma t_1 (floor h) (* (* (* dX.u (floor w)) dX.u) (floor w)))
           (fma (* dY.v dY.v) t_5 (* (* t_4 dY.u) (floor w))))))
        dX.u)
       (*
        (/
         (floor w)
         (sqrt
          (fmax
           (fma (* (* dX.u dX.u) (floor w)) (floor w) t_2)
           (fma t_3 (floor h) (* (* t_4 (floor w)) dY.u)))))
        dY.u))
     (if (>= t_2 t_6) (/ (* (floor w) dX.u) t_7) (/ (* (floor w) dY.u) 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 = floorf(w) * floorf(w);
	float t_1 = (floorf(h) * dX_46_v) * dX_46_v;
	float t_2 = t_1 * floorf(h);
	float t_3 = (floorf(h) * dY_46_v) * dY_46_v;
	float t_4 = dY_46_u * floorf(w);
	float t_5 = floorf(h) * floorf(h);
	float t_6 = fmaf((t_0 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_5));
	float t_7 = sqrtf(fmaxf(fmaf((t_0 * dX_46_u), dX_46_u, (t_5 * (dX_46_v * dX_46_v))), t_6));
	float tmp_1;
	if (dX_46_v <= 0.029999999329447746f) {
		float tmp_2;
		if (((dX_46_u * dX_46_u) * t_0) >= fmaf(t_3, floorf(h), ((dY_46_u * dY_46_u) * t_0))) {
			tmp_2 = (floorf(w) / sqrtf(fmaxf(fmaf(t_1, floorf(h), (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))), fmaf((dY_46_v * dY_46_v), t_5, ((t_4 * dY_46_u) * floorf(w)))))) * dX_46_u;
		} else {
			tmp_2 = (floorf(w) / sqrtf(fmaxf(fmaf(((dX_46_u * dX_46_u) * floorf(w)), floorf(w), t_2), fmaf(t_3, floorf(h), ((t_4 * floorf(w)) * dY_46_u))))) * dY_46_u;
		}
		tmp_1 = tmp_2;
	} else if (t_2 >= t_6) {
		tmp_1 = (floorf(w) * dX_46_u) / t_7;
	} else {
		tmp_1 = (floorf(w) * dY_46_u) / 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(floor(w) * floor(w))
	t_1 = Float32(Float32(floor(h) * dX_46_v) * dX_46_v)
	t_2 = Float32(t_1 * floor(h))
	t_3 = Float32(Float32(floor(h) * dY_46_v) * dY_46_v)
	t_4 = Float32(dY_46_u * floor(w))
	t_5 = Float32(floor(h) * floor(h))
	t_6 = fma(Float32(t_0 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_5))
	t_7 = sqrt(fmax(fma(Float32(t_0 * dX_46_u), dX_46_u, Float32(t_5 * Float32(dX_46_v * dX_46_v))), t_6))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(0.029999999329447746))
		tmp_2 = Float32(0.0)
		if (Float32(Float32(dX_46_u * dX_46_u) * t_0) >= fma(t_3, floor(h), Float32(Float32(dY_46_u * dY_46_u) * t_0)))
			tmp_2 = Float32(Float32(floor(w) / sqrt(fmax(fma(t_1, floor(h), Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))), fma(Float32(dY_46_v * dY_46_v), t_5, Float32(Float32(t_4 * dY_46_u) * floor(w)))))) * dX_46_u);
		else
			tmp_2 = Float32(Float32(floor(w) / sqrt(fmax(fma(Float32(Float32(dX_46_u * dX_46_u) * floor(w)), floor(w), t_2), fma(t_3, floor(h), Float32(Float32(t_4 * floor(w)) * dY_46_u))))) * dY_46_u);
		end
		tmp_1 = tmp_2;
	elseif (t_2 >= t_6)
		tmp_1 = Float32(Float32(floor(w) * dX_46_u) / t_7);
	else
		tmp_1 = Float32(Float32(floor(w) * dY_46_u) / t_7);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.0299999993

   1. Initial program 77.5%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 69.5

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

   4. Applied rewrites 69.5%

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

   6. Applied rewrites 69.3%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f32 N/A

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

      6. lower-*.f32 69.3

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 69.3

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 69.3

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

   8. Applied rewrites 69.3%

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

   10. Applied rewrites 69.4%

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

   if 0.0299999993 < dX.v

   1. Initial program 72.9%

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

   3. Applied rewrites 73.0%

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

   4. Taylor expanded in dX.u around 0

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

      1. lift-floor.f32 N/A

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

      2. lift-floor.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 N/A

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

      8. swap-sqr N/A

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

      9. swap-sqr N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   6. Applied rewrites 69.4%

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

Alternative 6: 64.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* (floor h) dY.v) dY.v))
        (t_1 (* dY.u (floor w)))
        (t_2 (* (floor w) (floor w)))
        (t_3 (* (* (floor h) dX.v) dX.v)))
   (if (>= (* (* dX.u dX.u) t_2) (fma t_0 (floor h) (* (* dY.u dY.u) t_2)))
     (*
      (/
       (floor w)
       (sqrt
        (fmax
         (fma t_3 (floor h) (* (* (* dX.u (floor w)) dX.u) (floor w)))
         (fma
          (* dY.v dY.v)
          (* (floor h) (floor h))
          (* (* t_1 dY.u) (floor w))))))
      dX.u)
     (*
      (/
       (floor w)
       (sqrt
        (fmax
         (fma (* (* dX.u dX.u) (floor w)) (floor w) (* t_3 (floor h)))
         (fma t_0 (floor h) (* (* t_1 (floor w)) dY.u)))))
      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(h) * dY_46_v) * dY_46_v;
	float t_1 = dY_46_u * floorf(w);
	float t_2 = floorf(w) * floorf(w);
	float t_3 = (floorf(h) * dX_46_v) * dX_46_v;
	float tmp;
	if (((dX_46_u * dX_46_u) * t_2) >= fmaf(t_0, floorf(h), ((dY_46_u * dY_46_u) * t_2))) {
		tmp = (floorf(w) / sqrtf(fmaxf(fmaf(t_3, floorf(h), (((dX_46_u * floorf(w)) * dX_46_u) * floorf(w))), fmaf((dY_46_v * dY_46_v), (floorf(h) * floorf(h)), ((t_1 * dY_46_u) * floorf(w)))))) * dX_46_u;
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(fmaf(((dX_46_u * dX_46_u) * floorf(w)), floorf(w), (t_3 * floorf(h))), fmaf(t_0, floorf(h), ((t_1 * floorf(w)) * dY_46_u))))) * 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(Float32(floor(h) * dY_46_v) * dY_46_v)
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(floor(w) * floor(w))
	t_3 = Float32(Float32(floor(h) * dX_46_v) * dX_46_v)
	tmp = Float32(0.0)
	if (Float32(Float32(dX_46_u * dX_46_u) * t_2) >= fma(t_0, floor(h), Float32(Float32(dY_46_u * dY_46_u) * t_2)))
		tmp = Float32(Float32(floor(w) / sqrt(fmax(fma(t_3, floor(h), Float32(Float32(Float32(dX_46_u * floor(w)) * dX_46_u) * floor(w))), fma(Float32(dY_46_v * dY_46_v), Float32(floor(h) * floor(h)), Float32(Float32(t_1 * dY_46_u) * floor(w)))))) * dX_46_u);
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(fma(Float32(Float32(dX_46_u * dX_46_u) * floor(w)), floor(w), Float32(t_3 * floor(h))), fma(t_0, floor(h), Float32(Float32(t_1 * floor(w)) * dY_46_u))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in dX.u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.7

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

4. Applied rewrites 64.7%

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

6. Applied rewrites 64.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 64.6

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 64.6

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 64.6

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

8. Applied rewrites 64.6%

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

10. Applied rewrites 64.6%

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

Alternative 7: 64.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (* (floor h) dY.v) dY.v))
        (t_1 (* (floor w) (floor w)))
        (t_2 (fma t_0 (floor h) (* (* dY.u dY.u) t_1)))
        (t_3
         (fma
          (* (* dX.u dX.u) (floor w))
          (floor w)
          (* (* (* (floor h) dX.v) dX.v) (floor h)))))
   (if (>= (* (* dX.u dX.u) t_1) t_2)
     (* (/ (floor w) (sqrt (fmax t_3 t_2))) dX.u)
     (*
      (/
       (floor w)
       (sqrt
        (fmax
         t_3
         (fma t_0 (floor h) (* (* (* dY.u (floor w)) (floor w)) dY.u)))))
      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(h) * dY_46_v) * dY_46_v;
	float t_1 = floorf(w) * floorf(w);
	float t_2 = fmaf(t_0, floorf(h), ((dY_46_u * dY_46_u) * t_1));
	float t_3 = fmaf(((dX_46_u * dX_46_u) * floorf(w)), floorf(w), (((floorf(h) * dX_46_v) * dX_46_v) * floorf(h)));
	float tmp;
	if (((dX_46_u * dX_46_u) * t_1) >= t_2) {
		tmp = (floorf(w) / sqrtf(fmaxf(t_3, t_2))) * dX_46_u;
	} else {
		tmp = (floorf(w) / sqrtf(fmaxf(t_3, fmaf(t_0, floorf(h), (((dY_46_u * floorf(w)) * floorf(w)) * dY_46_u))))) * 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(Float32(floor(h) * dY_46_v) * dY_46_v)
	t_1 = Float32(floor(w) * floor(w))
	t_2 = fma(t_0, floor(h), Float32(Float32(dY_46_u * dY_46_u) * t_1))
	t_3 = fma(Float32(Float32(dX_46_u * dX_46_u) * floor(w)), floor(w), Float32(Float32(Float32(floor(h) * dX_46_v) * dX_46_v) * floor(h)))
	tmp = Float32(0.0)
	if (Float32(Float32(dX_46_u * dX_46_u) * t_1) >= t_2)
		tmp = Float32(Float32(floor(w) / sqrt(fmax(t_3, t_2))) * dX_46_u);
	else
		tmp = Float32(Float32(floor(w) / sqrt(fmax(t_3, fma(t_0, floor(h), Float32(Float32(Float32(dY_46_u * floor(w)) * floor(w)) * dY_46_u))))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in dX.u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.7

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

4. Applied rewrites 64.7%

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

6. Applied rewrites 64.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 64.6

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 64.6

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 64.6

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

8. Applied rewrites 64.6%

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

Alternative 8: 59.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) (floor w)))
        (t_1 (fma (* (* (floor h) dY.v) dY.v) (floor h) (* (* dY.u dY.u) t_0)))
        (t_2
         (/
          (floor w)
          (sqrt
           (fmax
            (fma
             (* (* dX.u dX.u) (floor w))
             (floor w)
             (* (* (* (floor h) dX.v) dX.v) (floor h)))
            t_1)))))
   (if (>= (* (* dX.u dX.u) t_0) t_1) (* t_2 dX.u) (* t_2 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 = fmaf(((floorf(h) * dY_46_v) * dY_46_v), floorf(h), ((dY_46_u * dY_46_u) * t_0));
	float t_2 = floorf(w) / sqrtf(fmaxf(fmaf(((dX_46_u * dX_46_u) * floorf(w)), floorf(w), (((floorf(h) * dX_46_v) * dX_46_v) * floorf(h))), t_1));
	float tmp;
	if (((dX_46_u * dX_46_u) * t_0) >= t_1) {
		tmp = t_2 * dX_46_u;
	} else {
		tmp = t_2 * 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 = fma(Float32(Float32(floor(h) * dY_46_v) * dY_46_v), floor(h), Float32(Float32(dY_46_u * dY_46_u) * t_0))
	t_2 = Float32(floor(w) / sqrt(fmax(fma(Float32(Float32(dX_46_u * dX_46_u) * floor(w)), floor(w), Float32(Float32(Float32(floor(h) * dX_46_v) * dX_46_v) * floor(h))), t_1)))
	tmp = Float32(0.0)
	if (Float32(Float32(dX_46_u * dX_46_u) * t_0) >= t_1)
		tmp = Float32(t_2 * dX_46_u);
	else
		tmp = Float32(t_2 * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in dX.u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.7

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

4. Applied rewrites 64.7%

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

6. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2025110 
(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))))