Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 75.6% → 75.7%

Time: 9.7s

Alternatives: 10

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: 75.6% 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: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = (dX_46_v * floorf(h)) * dX_46_v;
	float t_1 = dY_46_u * floorf(w);
	float t_2 = (dX_46_u * floorf(w)) * dX_46_u;
	float t_3 = dY_46_v * floorf(h);
	float t_4 = fmaf((t_1 * dY_46_u), floorf(w), ((t_3 * dY_46_v) * floorf(h)));
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = floorf(h) * dX_46_v;
	float tmp;
	if (fmaf(t_2, floorf(w), (t_0 * floorf(h))) >= t_4) {
		tmp = ((dX_46_u * 1.0f) / sqrtf(fmaxf(fmaf(t_0, floorf(h), (t_2 * floorf(w))), t_4))) * floorf(w);
	} else {
		tmp = (-dY_46_u * floorf(w)) / -sqrtf(fmaxf(fmaf(t_5, t_5, (t_6 * t_6)), fmaf(t_1, t_1, (t_3 * 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(Float32(dX_46_v * floor(h)) * dX_46_v)
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(Float32(dX_46_u * floor(w)) * dX_46_u)
	t_3 = Float32(dY_46_v * floor(h))
	t_4 = fma(Float32(t_1 * dY_46_u), floor(w), Float32(Float32(t_3 * dY_46_v) * floor(h)))
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (fma(t_2, floor(w), Float32(t_0 * floor(h))) >= t_4)
		tmp = Float32(Float32(Float32(dX_46_u * Float32(1.0)) / sqrt(fmax(fma(t_0, floor(h), Float32(t_2 * floor(w))), t_4))) * floor(w));
	else
		tmp = Float32(Float32(Float32(-dY_46_u) * floor(w)) / Float32(-sqrt(fmax(fma(t_5, t_5, Float32(t_6 * t_6)), fma(t_1, t_1, Float32(t_3 * t_3))))));
	end
	return tmp
end

Derivation

1. Initial program 75.6%

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

3. Applied rewrites 75.5%

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

5. Applied rewrites 75.6%

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

7. Applied rewrites 75.6%

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

9. Applied rewrites 75.7%

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

Alternative 2: 75.7% 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 75.6%

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

3. Applied rewrites 75.7%

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

Alternative 3: 68.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_2 := \left(dY.v \cdot dY.v\right) \cdot t\_1\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_6 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_7 := t\_3 \cdot t\_3 + t\_4 \cdot t\_4\\ t_8 := \frac{1}{\sqrt{\mathsf{max}\left(t\_7, t\_0 \cdot t\_0 + t\_6 \cdot t\_6\right)}}\\ t_9 := t\_8 \cdot t\_0\\ \mathbf{if}\;dY.v \leq 0.0008200000156648457:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_7 \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_5 \cdot dX.u, dX.u, t\_1 \cdot \left(dX.v \cdot dX.v\right)\right), \mathsf{fma}\left(t\_5 \cdot dY.u, dY.u, t\_2\right)\right)}}\right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;t\_9\\ \end{array}\\ \mathbf{elif}\;t\_7 \geq t\_2:\\ \;\;\;\;t\_8 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_9\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) (floor h)))
        (t_2 (* (* dY.v dY.v) t_1))
        (t_3 (* (floor w) dX.u))
        (t_4 (* (floor h) dX.v))
        (t_5 (* (floor w) (floor w)))
        (t_6 (* (floor h) dY.v))
        (t_7 (+ (* t_3 t_3) (* t_4 t_4)))
        (t_8 (/ 1.0 (sqrt (fmax t_7 (+ (* t_0 t_0) (* t_6 t_6))))))
        (t_9 (* t_8 t_0)))
   (if (<= dY.v 0.0008200000156648457)
     (if (>= t_7 (* (* (* dY.u (floor w)) dY.u) (floor w)))
       (*
        (*
         (floor w)
         (sqrt
          (/
           1.0
           (fmax
            (fma (* t_5 dX.u) dX.u (* t_1 (* dX.v dX.v)))
            (fma (* t_5 dY.u) dY.u t_2)))))
        dX.u)
       t_9)
     (if (>= t_7 t_2) (* t_8 t_3) t_9))))

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) * dY_46_u;
	float t_1 = floorf(h) * floorf(h);
	float t_2 = (dY_46_v * dY_46_v) * t_1;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(w) * floorf(w);
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = (t_3 * t_3) + (t_4 * t_4);
	float t_8 = 1.0f / sqrtf(fmaxf(t_7, ((t_0 * t_0) + (t_6 * t_6))));
	float t_9 = t_8 * t_0;
	float tmp_1;
	if (dY_46_v <= 0.0008200000156648457f) {
		float tmp_2;
		if (t_7 >= (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w))) {
			tmp_2 = (floorf(w) * sqrtf((1.0f / fmaxf(fmaf((t_5 * dX_46_u), dX_46_u, (t_1 * (dX_46_v * dX_46_v))), fmaf((t_5 * dY_46_u), dY_46_u, t_2))))) * dX_46_u;
		} else {
			tmp_2 = t_9;
		}
		tmp_1 = tmp_2;
	} else if (t_7 >= t_2) {
		tmp_1 = t_8 * t_3;
	} else {
		tmp_1 = t_9;
	}
	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) * dY_46_u)
	t_1 = Float32(floor(h) * floor(h))
	t_2 = Float32(Float32(dY_46_v * dY_46_v) * t_1)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(w) * floor(w))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_3 * t_3) + Float32(t_4 * t_4))
	t_8 = Float32(Float32(1.0) / sqrt(fmax(t_7, Float32(Float32(t_0 * t_0) + Float32(t_6 * t_6)))))
	t_9 = Float32(t_8 * t_0)
	tmp_1 = Float32(0.0)
	if (dY_46_v <= Float32(0.0008200000156648457))
		tmp_2 = Float32(0.0)
		if (t_7 >= Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w)))
			tmp_2 = Float32(Float32(floor(w) * sqrt(Float32(Float32(1.0) / fmax(fma(Float32(t_5 * dX_46_u), dX_46_u, Float32(t_1 * Float32(dX_46_v * dX_46_v))), fma(Float32(t_5 * dY_46_u), dY_46_u, t_2))))) * dX_46_u);
		else
			tmp_2 = t_9;
		end
		tmp_1 = tmp_2;
	elseif (t_7 >= t_2)
		tmp_1 = Float32(t_8 * t_3);
	else
		tmp_1 = t_9;
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 8.20000016e-4

   1. Initial program 77.1%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in dY.u around inf

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

      1. lift-floor.f32 N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 68.3

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 68.3

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

   6. Applied rewrites 68.3%

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

   if 8.20000016e-4 < dY.v

   1. Initial program 72.0%

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

   2. Taylor expanded in dY.u around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 67.0

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

   4. Applied rewrites 67.0%

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

Alternative 4: 67.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \mathsf{fma}\left(t\_0 \cdot dY.u, \left\lfloor w\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_6 := t\_3 \cdot t\_3 + t\_5 \cdot t\_5\\ t_7 := \left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v\\ t_8 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_9 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_10 := \left(t\_9 \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor \\ t_11 := \sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_7, \left\lfloor h\right\rfloor , t\_10\right), t\_1\right)}\\ t_12 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;dX.v \leq 10:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_10 \geq t\_6:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_2 \cdot t\_2 + t\_4 \cdot t\_4, t\_6\right)}} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left(t\_8 \cdot dX.u, dX.u, t\_12 \cdot \left(dX.v \cdot dX.v\right)\right), \mathsf{fma}\left(t\_8 \cdot dY.u, dY.u, \left(dY.v \cdot dY.v\right) \cdot t\_12\right)\right)}}\right) \cdot dY.u\\ \end{array}\\ \mathbf{elif}\;t\_7 \cdot \left\lfloor h\right\rfloor \geq t\_1:\\ \;\;\;\;\frac{t\_9}{t\_11}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_11}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.u (floor w)))
        (t_1
         (fma
          (* t_0 dY.u)
          (floor w)
          (* (* (* dY.v (floor h)) dY.v) (floor h))))
        (t_2 (* (floor w) dX.u))
        (t_3 (* (floor w) dY.u))
        (t_4 (* (floor h) dX.v))
        (t_5 (* (floor h) dY.v))
        (t_6 (+ (* t_3 t_3) (* t_5 t_5)))
        (t_7 (* (* dX.v (floor h)) dX.v))
        (t_8 (* (floor w) (floor w)))
        (t_9 (* dX.u (floor w)))
        (t_10 (* (* t_9 dX.u) (floor w)))
        (t_11 (sqrt (fmax (fma t_7 (floor h) t_10) t_1)))
        (t_12 (* (floor h) (floor h))))
   (if (<= dX.v 10.0)
     (if (>= t_10 t_6)
       (* (/ 1.0 (sqrt (fmax (+ (* t_2 t_2) (* t_4 t_4)) t_6))) t_2)
       (*
        (*
         (floor w)
         (sqrt
          (/
           1.0
           (fmax
            (fma (* t_8 dX.u) dX.u (* t_12 (* dX.v dX.v)))
            (fma (* t_8 dY.u) dY.u (* (* dY.v dY.v) t_12))))))
        dY.u))
     (if (>= (* t_7 (floor h)) t_1) (/ t_9 t_11) (/ t_0 t_11)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dY_46_u * floorf(w);
	float t_1 = fmaf((t_0 * dY_46_u), floorf(w), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)));
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = (t_3 * t_3) + (t_5 * t_5);
	float t_7 = (dX_46_v * floorf(h)) * dX_46_v;
	float t_8 = floorf(w) * floorf(w);
	float t_9 = dX_46_u * floorf(w);
	float t_10 = (t_9 * dX_46_u) * floorf(w);
	float t_11 = sqrtf(fmaxf(fmaf(t_7, floorf(h), t_10), t_1));
	float t_12 = floorf(h) * floorf(h);
	float tmp_1;
	if (dX_46_v <= 10.0f) {
		float tmp_2;
		if (t_10 >= t_6) {
			tmp_2 = (1.0f / sqrtf(fmaxf(((t_2 * t_2) + (t_4 * t_4)), t_6))) * t_2;
		} else {
			tmp_2 = (floorf(w) * sqrtf((1.0f / fmaxf(fmaf((t_8 * dX_46_u), dX_46_u, (t_12 * (dX_46_v * dX_46_v))), fmaf((t_8 * dY_46_u), dY_46_u, ((dY_46_v * dY_46_v) * t_12)))))) * dY_46_u;
		}
		tmp_1 = tmp_2;
	} else if ((t_7 * floorf(h)) >= t_1) {
		tmp_1 = t_9 / t_11;
	} else {
		tmp_1 = t_0 / t_11;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_u * floor(w))
	t_1 = fma(Float32(t_0 * dY_46_u), floor(w), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(Float32(t_3 * t_3) + Float32(t_5 * t_5))
	t_7 = Float32(Float32(dX_46_v * floor(h)) * dX_46_v)
	t_8 = Float32(floor(w) * floor(w))
	t_9 = Float32(dX_46_u * floor(w))
	t_10 = Float32(Float32(t_9 * dX_46_u) * floor(w))
	t_11 = sqrt(fmax(fma(t_7, floor(h), t_10), t_1))
	t_12 = Float32(floor(h) * floor(h))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(10.0))
		tmp_2 = Float32(0.0)
		if (t_10 >= t_6)
			tmp_2 = Float32(Float32(Float32(1.0) / sqrt(fmax(Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)), t_6))) * t_2);
		else
			tmp_2 = Float32(Float32(floor(w) * sqrt(Float32(Float32(1.0) / fmax(fma(Float32(t_8 * dX_46_u), dX_46_u, Float32(t_12 * Float32(dX_46_v * dX_46_v))), fma(Float32(t_8 * dY_46_u), dY_46_u, Float32(Float32(dY_46_v * dY_46_v) * t_12)))))) * dY_46_u);
		end
		tmp_1 = tmp_2;
	elseif (Float32(t_7 * floor(h)) >= t_1)
		tmp_1 = Float32(t_9 / t_11);
	else
		tmp_1 = Float32(t_0 / t_11);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 10

   1. Initial program 77.4%

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in dX.u around inf

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

      1. lift-floor.f32 N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 69.3

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 69.3

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

   6. Applied rewrites 69.3%

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

   if 10 < dX.v

   1. Initial program 70.0%

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

   2. Taylor expanded in dX.u around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 66.5

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

   4. Applied rewrites 66.5%

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

   6. Applied rewrites 66.6%

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

Alternative 5: 64.7% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.6%

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

2. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.5

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

4. Applied rewrites 64.5%

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

6. Applied rewrites 64.7%

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

Alternative 6: 60.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := t\_0 \cdot t\_0 + t\_1 \cdot t\_1\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := t\_4 \cdot t\_4\\ \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_3 \cdot t\_3 + t\_5, t\_2\right)}} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(e^{\log \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot 2} + t\_5, t\_2\right)}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (+ (* t_0 t_0) (* t_1 t_1)))
        (t_3 (* (floor w) dX.u))
        (t_4 (* (floor h) dX.v))
        (t_5 (* t_4 t_4)))
   (if (>=
        (* (* (floor h) (floor h)) (* dX.v dX.v))
        (* (* (* dY.u (floor w)) dY.u) (floor w)))
     (* (/ 1.0 (sqrt (fmax (+ (* t_3 t_3) t_5) t_2))) t_3)
     (*
      (/ 1.0 (sqrt (fmax (+ (exp (* (log (* dX.u (floor w))) 2.0)) t_5) t_2)))
      t_0))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = (t_0 * t_0) + (t_1 * t_1);
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = t_4 * t_4;
	float tmp;
	if (((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v)) >= (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w))) {
		tmp = (1.0f / sqrtf(fmaxf(((t_3 * t_3) + t_5), t_2))) * t_3;
	} else {
		tmp = (1.0f / sqrtf(fmaxf((expf((logf((dX_46_u * floorf(w))) * 2.0f)) + t_5), t_2))) * t_0;
	}
	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) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(t_4 * t_4)
	tmp = Float32(0.0)
	if (Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)) >= Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w)))
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(Float32(Float32(t_3 * t_3) + t_5), t_2))) * t_3);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(Float32(exp(Float32(log(Float32(dX_46_u * floor(w))) * Float32(2.0))) + t_5), t_2))) * t_0);
	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(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = (t_0 * t_0) + (t_1 * t_1);
	t_3 = floor(w) * dX_46_u;
	t_4 = floor(h) * dX_46_v;
	t_5 = t_4 * t_4;
	tmp = single(0.0);
	if (((floor(h) * floor(h)) * (dX_46_v * dX_46_v)) >= (((dY_46_u * floor(w)) * dY_46_u) * floor(w)))
		tmp = (single(1.0) / sqrt(max(((t_3 * t_3) + t_5), t_2))) * t_3;
	else
		tmp = (single(1.0) / sqrt(max((exp((log((dX_46_u * floor(w))) * single(2.0))) + t_5), t_2))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.6%

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

2. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in dY.u around inf

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

   1. lift-floor.f32 N/A

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

   2. pow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 58.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 58.9

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

7. Applied rewrites 58.9%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. unpow-prod-down N/A

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

   5. pow-to-exp N/A

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

   6. lift-log.f32 N/A

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

   7. pow-to-exp N/A

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

   8. exp-sum N/A

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

   9. lower-fma.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-exp.f32 60.5

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

   13. lift-fma.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-log.f32 N/A

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

   16. distribute-rgt-out N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

9. Applied rewrites 60.5%

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

Alternative 7: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_5 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_6 := t\_5 \cdot t\_5\\ \mathbf{if}\;\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right) \geq \left(t\_0 \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor :\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_4 \cdot t\_4 + t\_6, e^{\log t\_0 \cdot 2} + t\_3\right)}} \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(e^{\log \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot 2} + t\_6, t\_1 \cdot t\_1 + t\_3\right)}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.u (floor w)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* t_2 t_2))
        (t_4 (* (floor w) dX.u))
        (t_5 (* (floor h) dX.v))
        (t_6 (* t_5 t_5)))
   (if (>=
        (* (* (floor h) (floor h)) (* dX.v dX.v))
        (* (* t_0 dY.u) (floor w)))
     (*
      (/ 1.0 (sqrt (fmax (+ (* t_4 t_4) t_6) (+ (exp (* (log t_0) 2.0)) t_3))))
      t_4)
     (*
      (/
       1.0
       (sqrt
        (fmax
         (+ (exp (* (log (* dX.u (floor w))) 2.0)) t_6)
         (+ (* t_1 t_1) t_3))))
      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 = dY_46_u * floorf(w);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = floorf(h) * dX_46_v;
	float t_6 = t_5 * t_5;
	float tmp;
	if (((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v)) >= ((t_0 * dY_46_u) * floorf(w))) {
		tmp = (1.0f / sqrtf(fmaxf(((t_4 * t_4) + t_6), (expf((logf(t_0) * 2.0f)) + t_3)))) * t_4;
	} else {
		tmp = (1.0f / sqrtf(fmaxf((expf((logf((dX_46_u * floorf(w))) * 2.0f)) + t_6), ((t_1 * t_1) + t_3)))) * 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(dY_46_u * floor(w))
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(floor(h) * dX_46_v)
	t_6 = Float32(t_5 * t_5)
	tmp = Float32(0.0)
	if (Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)) >= Float32(Float32(t_0 * dY_46_u) * floor(w)))
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(Float32(Float32(t_4 * t_4) + t_6), Float32(exp(Float32(log(t_0) * Float32(2.0))) + t_3)))) * t_4);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(Float32(exp(Float32(log(Float32(dX_46_u * floor(w))) * Float32(2.0))) + t_6), Float32(Float32(t_1 * t_1) + t_3)))) * 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 = dY_46_u * floor(w);
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = t_2 * t_2;
	t_4 = floor(w) * dX_46_u;
	t_5 = floor(h) * dX_46_v;
	t_6 = t_5 * t_5;
	tmp = single(0.0);
	if (((floor(h) * floor(h)) * (dX_46_v * dX_46_v)) >= ((t_0 * dY_46_u) * floor(w)))
		tmp = (single(1.0) / sqrt(max(((t_4 * t_4) + t_6), (exp((log(t_0) * single(2.0))) + t_3)))) * t_4;
	else
		tmp = (single(1.0) / sqrt(max((exp((log((dX_46_u * floor(w))) * single(2.0))) + t_6), ((t_1 * t_1) + t_3)))) * t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 75.6%

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

2. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in dY.u around inf

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

   1. lift-floor.f32 N/A

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

   2. pow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 58.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 58.9

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

7. Applied rewrites 58.9%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. unpow-prod-down N/A

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

   5. pow-to-exp N/A

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

   6. lift-log.f32 N/A

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

   7. pow-to-exp N/A

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

   8. exp-sum N/A

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

   9. lower-fma.f32 N/A

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

   10. lift-log.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-exp.f32 60.5

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

   13. lift-fma.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-log.f32 N/A

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

   16. distribute-rgt-out N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 N/A

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

9. Applied rewrites 60.5%

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

    1. lift-*.f32 N/A

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

    2. pow2 N/A

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

    3. pow-to-exp N/A

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

    4. lower-exp.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lower-log.f32 60.4

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lift-*.f32 60.4

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

11. Applied rewrites 60.4%

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

Alternative 8: 59.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 75.6%

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

2. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in dY.u around inf

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

   1. lift-floor.f32 N/A

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

   2. pow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 58.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 58.9

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

7. Applied rewrites 58.9%

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

9. Applied rewrites 59.1%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. +-commutative N/A

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

    4. lift-*.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f32 N/A

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

    9. associate-*l* N/A

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

    10. lift-*.f32 N/A

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

    11. *-commutative N/A

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

    12. associate-*r* N/A

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

    13. lift-*.f32 N/A

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

    14. *-commutative N/A

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

    15. lower-fma.f32 N/A

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

11. Applied rewrites 59.1%

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

Alternative 9: 59.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 75.6%

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

2. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in dY.u around inf

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

   1. lift-floor.f32 N/A

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

   2. pow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 58.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 58.9

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

7. Applied rewrites 58.9%

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

9. Applied rewrites 59.1%

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

    1. lift-fma.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*l* N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f32 N/A

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

    9. *-commutative N/A

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

    10. lift-*.f32 N/A

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

    11. lift-*.f32 N/A

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

    12. lift-*.f32 N/A

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

    13. lift-*.f32 N/A

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

    14. associate-*l* N/A

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

    15. lift-*.f32 N/A

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

    16. associate-*l* N/A

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

    17. lift-*.f32 N/A

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

11. Applied rewrites 59.1%

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

Alternative 10: 58.9% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 75.6%

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

2. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 64.5

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

4. Applied rewrites 64.5%

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

5. Taylor expanded in dY.u around inf

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

   1. lift-floor.f32 N/A

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

   2. pow2 N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 58.9

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 58.9

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

7. Applied rewrites 58.9%

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

9. Applied rewrites 59.1%

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

11. Applied rewrites 58.9%

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

Reproduce

herbie shell --seed 2025112 
(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))))