Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.1% → 76.4%

Time: 10.8s

Alternatives: 13

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

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) 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_0) (* t_6 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) * 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_0;
	} else {
		tmp = t_6 * 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) * 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_0);
	else
		tmp = Float32(t_6 * t_4);
	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_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) 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_0) (* t_6 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) * 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_0;
	} else {
		tmp = t_6 * 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) * 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_0);
	else
		tmp = Float32(t_6 * t_4);
	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_0;
	else
		tmp = t_6 * t_4;
	end
	tmp_2 = tmp;
end

Alternative 1: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

Alternative 2: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

5. Applied rewrites 76.2%

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

Alternative 3: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := {t\_2}^{2}\\ t_4 := t\_3 + t\_0\\ t_5 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_6 := {t\_5}^{2} + t\_1\\ t_7 := \sqrt{\mathsf{max}\left(t\_6, t\_0 + t\_3\right)}\\ \mathbf{if}\;dY.u \leq 8000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_6 \geq t\_3:\\ \;\;\;\;\frac{t\_5}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.v}{t\_7} \cdot \left\lfloor h\right\rfloor \\ \end{array}\\ \mathbf{elif}\;t\_6 \geq t\_0:\\ \;\;\;\;\frac{t\_5}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, t\_1\right), t\_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{max}\left(t\_6, t\_4\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dY.u) 2.0))
        (t_1 (pow (* (floor w) dX.u) 2.0))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow t_2 2.0))
        (t_4 (+ t_3 t_0))
        (t_5 (* (floor h) dX.v))
        (t_6 (+ (pow t_5 2.0) t_1))
        (t_7 (sqrt (fmax t_6 (+ t_0 t_3)))))
   (if (<= dY.u 8000000.0)
     (if (>= t_6 t_3) (/ t_5 t_7) (* (/ dY.v t_7) (floor h)))
     (if (>= t_6 t_0)
       (/ t_5 (sqrt (fmax (fma (* (pow (floor h) 2.0) dX.v) dX.v t_1) t_4)))
       (/ t_2 (sqrt (fmax t_6 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 = powf((floorf(w) * dY_46_u), 2.0f);
	float t_1 = powf((floorf(w) * dX_46_u), 2.0f);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = powf(t_2, 2.0f);
	float t_4 = t_3 + t_0;
	float t_5 = floorf(h) * dX_46_v;
	float t_6 = powf(t_5, 2.0f) + t_1;
	float t_7 = sqrtf(fmaxf(t_6, (t_0 + t_3)));
	float tmp_1;
	if (dY_46_u <= 8000000.0f) {
		float tmp_2;
		if (t_6 >= t_3) {
			tmp_2 = t_5 / t_7;
		} else {
			tmp_2 = (dY_46_v / t_7) * floorf(h);
		}
		tmp_1 = tmp_2;
	} else if (t_6 >= t_0) {
		tmp_1 = t_5 / sqrtf(fmaxf(fmaf((powf(floorf(h), 2.0f) * dX_46_v), dX_46_v, t_1), t_4));
	} else {
		tmp_1 = t_2 / sqrtf(fmaxf(t_6, t_4));
	}
	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) ^ Float32(2.0)
	t_1 = Float32(floor(w) * dX_46_u) ^ Float32(2.0)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = t_2 ^ Float32(2.0)
	t_4 = Float32(t_3 + t_0)
	t_5 = Float32(floor(h) * dX_46_v)
	t_6 = Float32((t_5 ^ Float32(2.0)) + t_1)
	t_7 = sqrt(fmax(t_6, Float32(t_0 + t_3)))
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(8000000.0))
		tmp_2 = Float32(0.0)
		if (t_6 >= t_3)
			tmp_2 = Float32(t_5 / t_7);
		else
			tmp_2 = Float32(Float32(dY_46_v / t_7) * floor(h));
		end
		tmp_1 = tmp_2;
	elseif (t_6 >= t_0)
		tmp_1 = Float32(t_5 / sqrt(fmax(fma(Float32((floor(h) ^ Float32(2.0)) * dX_46_v), dX_46_v, t_1), t_4)));
	else
		tmp_1 = Float32(t_2 / sqrt(fmax(t_6, t_4)));
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 8e6

   1. Initial program 78.0%

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

   2. Taylor expanded in dY.u around 0

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

      1. pow-prod-down N/A

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

      2. *-commutative N/A

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

      3. lift-floor.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lower-pow.f32 69.7

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

   4. Applied rewrites 69.7%

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

   6. Applied rewrites 69.8%

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

   if 8e6 < dY.u

   1. Initial program 66.2%

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

   3. Applied rewrites 66.4%

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

      1. lift-+.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-floor.f32 N/A

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

      5. unpow-prod-down N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lift-floor.f32 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in dY.u around inf

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

   8. Applied rewrites 64.8%

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

Alternative 4: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ t_2 := {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\\ t_3 := t\_2 + t\_0\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := {t\_4}^{2}\\ t_6 := t\_5 + t\_1\\ t_7 := \sqrt{\mathsf{max}\left(t\_6, t\_0 + t\_2\right)}\\ \mathbf{if}\;dY.u \leq 200000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_6 \geq t\_2:\\ \;\;\;\;\frac{t\_4}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.v}{t\_7} \cdot \left\lfloor h\right\rfloor \\ \end{array}\\ \mathbf{elif}\;t\_5 \geq t\_3:\\ \;\;\;\;\frac{t\_4}{\sqrt{\mathsf{max}\left(t\_6, t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{dY.v}{\sqrt{\mathsf{max}\left(t\_1 + t\_5, t\_3\right)}} \cdot \left\lfloor h\right\rfloor \\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dY.u) 2.0))
        (t_1 (pow (* (floor w) dX.u) 2.0))
        (t_2 (pow (* (floor h) dY.v) 2.0))
        (t_3 (+ t_2 t_0))
        (t_4 (* (floor h) dX.v))
        (t_5 (pow t_4 2.0))
        (t_6 (+ t_5 t_1))
        (t_7 (sqrt (fmax t_6 (+ t_0 t_2)))))
   (if (<= dY.u 200000.0)
     (if (>= t_6 t_2) (/ t_4 t_7) (* (/ dY.v t_7) (floor h)))
     (if (>= t_5 t_3)
       (/ t_4 (sqrt (fmax t_6 t_3)))
       (* (/ dY.v (sqrt (fmax (+ t_1 t_5) t_3))) (floor h))))))

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 = powf((floorf(w) * dY_46_u), 2.0f);
	float t_1 = powf((floorf(w) * dX_46_u), 2.0f);
	float t_2 = powf((floorf(h) * dY_46_v), 2.0f);
	float t_3 = t_2 + t_0;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = powf(t_4, 2.0f);
	float t_6 = t_5 + t_1;
	float t_7 = sqrtf(fmaxf(t_6, (t_0 + t_2)));
	float tmp_1;
	if (dY_46_u <= 200000.0f) {
		float tmp_2;
		if (t_6 >= t_2) {
			tmp_2 = t_4 / t_7;
		} else {
			tmp_2 = (dY_46_v / t_7) * floorf(h);
		}
		tmp_1 = tmp_2;
	} else if (t_5 >= t_3) {
		tmp_1 = t_4 / sqrtf(fmaxf(t_6, t_3));
	} else {
		tmp_1 = (dY_46_v / sqrtf(fmaxf((t_1 + t_5), t_3))) * floorf(h);
	}
	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) ^ Float32(2.0)
	t_1 = Float32(floor(w) * dX_46_u) ^ Float32(2.0)
	t_2 = Float32(floor(h) * dY_46_v) ^ Float32(2.0)
	t_3 = Float32(t_2 + t_0)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = t_4 ^ Float32(2.0)
	t_6 = Float32(t_5 + t_1)
	t_7 = sqrt(fmax(t_6, Float32(t_0 + t_2)))
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(200000.0))
		tmp_2 = Float32(0.0)
		if (t_6 >= t_2)
			tmp_2 = Float32(t_4 / t_7);
		else
			tmp_2 = Float32(Float32(dY_46_v / t_7) * floor(h));
		end
		tmp_1 = tmp_2;
	elseif (t_5 >= t_3)
		tmp_1 = Float32(t_4 / sqrt(fmax(t_6, t_3)));
	else
		tmp_1 = Float32(Float32(dY_46_v / sqrt(fmax(Float32(t_1 + t_5), t_3))) * floor(h));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = (floor(w) * dY_46_u) ^ single(2.0);
	t_1 = (floor(w) * dX_46_u) ^ single(2.0);
	t_2 = (floor(h) * dY_46_v) ^ single(2.0);
	t_3 = t_2 + t_0;
	t_4 = floor(h) * dX_46_v;
	t_5 = t_4 ^ single(2.0);
	t_6 = t_5 + t_1;
	t_7 = sqrt(max(t_6, (t_0 + t_2)));
	tmp_2 = single(0.0);
	if (dY_46_u <= single(200000.0))
		tmp_3 = single(0.0);
		if (t_6 >= t_2)
			tmp_3 = t_4 / t_7;
		else
			tmp_3 = (dY_46_v / t_7) * floor(h);
		end
		tmp_2 = tmp_3;
	elseif (t_5 >= t_3)
		tmp_2 = t_4 / sqrt(max(t_6, t_3));
	else
		tmp_2 = (dY_46_v / sqrt(max((t_1 + t_5), t_3))) * floor(h);
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2e5

   1. Initial program 77.9%

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

   2. Taylor expanded in dY.u around 0

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

      1. pow-prod-down N/A

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

      2. *-commutative N/A

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

      3. lift-floor.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lower-pow.f32 69.8

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

   4. Applied rewrites 69.8%

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

   6. Applied rewrites 69.9%

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

   if 2e5 < dY.u

   1. Initial program 67.9%

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

   3. Applied rewrites 68.1%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 61.4%

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

   8. Applied rewrites 61.3%

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

Alternative 5: 65.7% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

4. Taylor expanded in dX.u around 0

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

6. Applied rewrites 65.7%

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

Alternative 6: 65.5% accurate, 1.2× speedup

Math

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

FPCore

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

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 = powf(t_0, 2.0f);
	float t_2 = powf((floorf(w) * dX_46_u), 2.0f);
	float t_3 = powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(w) * dY_46_u), 2.0f);
	float tmp;
	if (t_1 >= t_3) {
		tmp = t_0 / sqrtf(fmaxf((t_1 + t_2), t_3));
	} else {
		tmp = (dY_46_v / sqrtf(fmaxf((t_2 + t_1), t_3))) * floorf(h);
	}
	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 = t_0 ^ Float32(2.0)
	t_2 = Float32(floor(w) * dX_46_u) ^ Float32(2.0)
	t_3 = Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (t_1 >= t_3)
		tmp = Float32(t_0 / sqrt(fmax(Float32(t_1 + t_2), t_3)));
	else
		tmp = Float32(Float32(dY_46_v / sqrt(fmax(Float32(t_2 + t_1), t_3))) * floor(h));
	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 = t_0 ^ single(2.0);
	t_2 = (floor(w) * dX_46_u) ^ single(2.0);
	t_3 = ((floor(h) * dY_46_v) ^ single(2.0)) + ((floor(w) * dY_46_u) ^ single(2.0));
	tmp = single(0.0);
	if (t_1 >= t_3)
		tmp = t_0 / sqrt(max((t_1 + t_2), t_3));
	else
		tmp = (dY_46_v / sqrt(max((t_2 + t_1), t_3))) * floor(h);
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

4. Taylor expanded in dX.u around 0

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

6. Applied rewrites 65.7%

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

8. Applied rewrites 65.5%

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

Alternative 7: 65.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = powf(t_0, 2.0f);
	float t_2 = powf((floorf(w) * dX_46_u), 2.0f);
	float t_3 = powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(w) * dY_46_u), 2.0f);
	float tmp;
	if (t_1 >= t_3) {
		tmp = t_0 / sqrtf(fmaxf((t_1 + t_2), t_3));
	} else {
		tmp = dY_46_v * (floorf(h) / sqrtf(fmaxf((t_2 + t_1), t_3)));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = t_0 ^ Float32(2.0)
	t_2 = Float32(floor(w) * dX_46_u) ^ Float32(2.0)
	t_3 = Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (t_1 >= t_3)
		tmp = Float32(t_0 / sqrt(fmax(Float32(t_1 + t_2), t_3)));
	else
		tmp = Float32(dY_46_v * Float32(floor(h) / sqrt(fmax(Float32(t_2 + t_1), t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = t_0 ^ single(2.0);
	t_2 = (floor(w) * dX_46_u) ^ single(2.0);
	t_3 = ((floor(h) * dY_46_v) ^ single(2.0)) + ((floor(w) * dY_46_u) ^ single(2.0));
	tmp = single(0.0);
	if (t_1 >= t_3)
		tmp = t_0 / sqrt(max((t_1 + t_2), t_3));
	else
		tmp = dY_46_v * (floor(h) / sqrt(max((t_2 + t_1), t_3)));
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

4. Taylor expanded in dX.u around 0

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

6. Applied rewrites 65.7%

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

8. Applied rewrites 65.5%

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

Alternative 8: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := {t\_0}^{2}\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := {t\_2}^{2}\\ t_4 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\\ t_5 := t\_3 + t\_4\\ t_6 := \sqrt{\mathsf{max}\left(t\_1 + {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, t\_5\right)}\\ t_7 := \frac{t\_2}{t\_6}\\ \mathbf{if}\;dY.u \leq 8000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_1 \geq t\_3:\\ \;\;\;\;\frac{t\_0}{\sqrt{\mathsf{max}\left(t\_1 + {\left(\frac{1}{{\left(\left\lfloor w\right\rfloor \right)}^{-1}} \cdot dX.u\right)}^{2}, t\_5\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array}\\ \mathbf{elif}\;t\_1 \geq t\_4:\\ \;\;\;\;\frac{t\_0}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (pow t_0 2.0))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow t_2 2.0))
        (t_4 (pow (* (floor w) dY.u) 2.0))
        (t_5 (+ t_3 t_4))
        (t_6 (sqrt (fmax (+ t_1 (pow (* (floor w) dX.u) 2.0)) t_5)))
        (t_7 (/ t_2 t_6)))
   (if (<= dY.u 8000000.0)
     (if (>= t_1 t_3)
       (/
        t_0
        (sqrt
         (fmax (+ t_1 (pow (* (/ 1.0 (pow (floor w) -1.0)) dX.u) 2.0)) t_5)))
       t_7)
     (if (>= t_1 t_4) (/ t_0 t_6) t_7))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = powf(t_0, 2.0f);
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = powf(t_2, 2.0f);
	float t_4 = powf((floorf(w) * dY_46_u), 2.0f);
	float t_5 = t_3 + t_4;
	float t_6 = sqrtf(fmaxf((t_1 + powf((floorf(w) * dX_46_u), 2.0f)), t_5));
	float t_7 = t_2 / t_6;
	float tmp_1;
	if (dY_46_u <= 8000000.0f) {
		float tmp_2;
		if (t_1 >= t_3) {
			tmp_2 = t_0 / sqrtf(fmaxf((t_1 + powf(((1.0f / powf(floorf(w), -1.0f)) * dX_46_u), 2.0f)), t_5));
		} else {
			tmp_2 = t_7;
		}
		tmp_1 = tmp_2;
	} else if (t_1 >= t_4) {
		tmp_1 = t_0 / t_6;
	} else {
		tmp_1 = t_7;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = t_0 ^ Float32(2.0)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = t_2 ^ Float32(2.0)
	t_4 = Float32(floor(w) * dY_46_u) ^ Float32(2.0)
	t_5 = Float32(t_3 + t_4)
	t_6 = sqrt(fmax(Float32(t_1 + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), t_5))
	t_7 = Float32(t_2 / t_6)
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(8000000.0))
		tmp_2 = Float32(0.0)
		if (t_1 >= t_3)
			tmp_2 = Float32(t_0 / sqrt(fmax(Float32(t_1 + (Float32(Float32(Float32(1.0) / (floor(w) ^ Float32(-1.0))) * dX_46_u) ^ Float32(2.0))), t_5)));
		else
			tmp_2 = t_7;
		end
		tmp_1 = tmp_2;
	elseif (t_1 >= t_4)
		tmp_1 = Float32(t_0 / t_6);
	else
		tmp_1 = t_7;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = t_0 ^ single(2.0);
	t_2 = floor(h) * dY_46_v;
	t_3 = t_2 ^ single(2.0);
	t_4 = (floor(w) * dY_46_u) ^ single(2.0);
	t_5 = t_3 + t_4;
	t_6 = sqrt(max((t_1 + ((floor(w) * dX_46_u) ^ single(2.0))), t_5));
	t_7 = t_2 / t_6;
	tmp_2 = single(0.0);
	if (dY_46_u <= single(8000000.0))
		tmp_3 = single(0.0);
		if (t_1 >= t_3)
			tmp_3 = t_0 / sqrt(max((t_1 + (((single(1.0) / (floor(w) ^ single(-1.0))) * dX_46_u) ^ single(2.0))), t_5));
		else
			tmp_3 = t_7;
		end
		tmp_2 = tmp_3;
	elseif (t_1 >= t_4)
		tmp_2 = t_0 / t_6;
	else
		tmp_2 = t_7;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 8e6

   1. Initial program 78.0%

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

   3. Applied rewrites 78.3%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 66.7%

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

   7. Taylor expanded in dY.u around 0

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

   9. Applied rewrites 63.3%

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

       1. lift-floor.f32 N/A

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

       2. unpow1 N/A

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

       3. metadata-eval N/A

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

       4. pow-neg N/A

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

       5. lower-/.f32 N/A

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

       6. lower-pow.f32 N/A

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

       7. lift-floor.f32 63.2

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

   11. Applied rewrites 63.2%

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

   if 8e6 < dY.u

   1. Initial program 66.2%

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in dY.u around inf

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

   9. Applied rewrites 59.2%

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

Alternative 9: 62.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := {t\_1}^{2}\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := {t\_3}^{2}\\ t_5 := t\_2 + t\_0\\ t_6 := \sqrt{\mathsf{max}\left(t\_4 + {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, t\_5\right)}\\ t_7 := \frac{t\_3}{t\_6}\\ \mathbf{if}\;dY.u \leq 8000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_4 \geq t\_2:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloor w\right\rfloor , \left\lfloor w\right\rfloor , t\_4\right), t\_5\right)}}\\ \end{array}\\ \mathbf{elif}\;t\_4 \geq t\_0:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_6}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dY.u) 2.0))
        (t_1 (* (floor h) dY.v))
        (t_2 (pow t_1 2.0))
        (t_3 (* (floor h) dX.v))
        (t_4 (pow t_3 2.0))
        (t_5 (+ t_2 t_0))
        (t_6 (sqrt (fmax (+ t_4 (pow (* (floor w) dX.u) 2.0)) t_5)))
        (t_7 (/ t_3 t_6)))
   (if (<= dY.u 8000000.0)
     (if (>= t_4 t_2)
       t_7
       (/
        t_1
        (sqrt (fmax (fma (* (* dX.u dX.u) (floor w)) (floor w) t_4) t_5))))
     (if (>= t_4 t_0) t_7 (/ t_1 t_6)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf((floorf(w) * dY_46_u), 2.0f);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = powf(t_1, 2.0f);
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = powf(t_3, 2.0f);
	float t_5 = t_2 + t_0;
	float t_6 = sqrtf(fmaxf((t_4 + powf((floorf(w) * dX_46_u), 2.0f)), t_5));
	float t_7 = t_3 / t_6;
	float tmp_1;
	if (dY_46_u <= 8000000.0f) {
		float tmp_2;
		if (t_4 >= t_2) {
			tmp_2 = t_7;
		} else {
			tmp_2 = t_1 / sqrtf(fmaxf(fmaf(((dX_46_u * dX_46_u) * floorf(w)), floorf(w), t_4), t_5));
		}
		tmp_1 = tmp_2;
	} else if (t_4 >= t_0) {
		tmp_1 = t_7;
	} else {
		tmp_1 = t_1 / t_6;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dY_46_u) ^ Float32(2.0)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = t_3 ^ Float32(2.0)
	t_5 = Float32(t_2 + t_0)
	t_6 = sqrt(fmax(Float32(t_4 + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), t_5))
	t_7 = Float32(t_3 / t_6)
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(8000000.0))
		tmp_2 = Float32(0.0)
		if (t_4 >= t_2)
			tmp_2 = t_7;
		else
			tmp_2 = Float32(t_1 / sqrt(fmax(fma(Float32(Float32(dX_46_u * dX_46_u) * floor(w)), floor(w), t_4), t_5)));
		end
		tmp_1 = tmp_2;
	elseif (t_4 >= t_0)
		tmp_1 = t_7;
	else
		tmp_1 = Float32(t_1 / t_6);
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 8e6

   1. Initial program 78.0%

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

   3. Applied rewrites 78.3%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 66.7%

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

   7. Taylor expanded in dY.u around 0

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

   9. Applied rewrites 63.3%

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

       1. lift-+.f32 N/A

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

       2. lift-pow.f32 N/A

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

       3. lift-*.f32 N/A

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

       4. lift-floor.f32 N/A

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

       5. unpow-prod-down N/A

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

       6. *-commutative N/A

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

       7. lift-pow.f32 N/A

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

       8. lift-*.f32 N/A

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

       9. lift-floor.f32 N/A

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

       10. unpow-prod-down N/A

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

       11. *-commutative N/A

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

       12. +-commutative N/A

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

       13. fp-cancel-sign-sub-inv N/A

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

       14. *-commutative N/A

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

       15. unpow-prod-down N/A

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

       16. lift-floor.f32 N/A

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

       17. lift-*.f32 N/A

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

       18. lift-pow.f32 N/A

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

       19. fp-cancel-sign-sub-inv N/A

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

   11. Applied rewrites 63.3%

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

   if 8e6 < dY.u

   1. Initial program 66.2%

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in dY.u around inf

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

   9. Applied rewrites 59.2%

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

Alternative 10: 62.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := {t\_1}^{2}\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := {t\_3}^{2}\\ t_5 := \sqrt{\mathsf{max}\left(t\_4 + {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, t\_2 + t\_0\right)}\\ t_6 := \frac{t\_1}{t\_5}\\ t_7 := \frac{t\_3}{t\_5}\\ \mathbf{if}\;dY.u \leq 8000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_4 \geq t\_2:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array}\\ \mathbf{elif}\;t\_4 \geq t\_0:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dY.u) 2.0))
        (t_1 (* (floor h) dY.v))
        (t_2 (pow t_1 2.0))
        (t_3 (* (floor h) dX.v))
        (t_4 (pow t_3 2.0))
        (t_5 (sqrt (fmax (+ t_4 (pow (* (floor w) dX.u) 2.0)) (+ t_2 t_0))))
        (t_6 (/ t_1 t_5))
        (t_7 (/ t_3 t_5)))
   (if (<= dY.u 8000000.0)
     (if (>= t_4 t_2) t_7 t_6)
     (if (>= t_4 t_0) t_7 t_6))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = powf((floorf(w) * dY_46_u), 2.0f);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = powf(t_1, 2.0f);
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = powf(t_3, 2.0f);
	float t_5 = sqrtf(fmaxf((t_4 + powf((floorf(w) * dX_46_u), 2.0f)), (t_2 + t_0)));
	float t_6 = t_1 / t_5;
	float t_7 = t_3 / t_5;
	float tmp_1;
	if (dY_46_u <= 8000000.0f) {
		float tmp_2;
		if (t_4 >= t_2) {
			tmp_2 = t_7;
		} else {
			tmp_2 = t_6;
		}
		tmp_1 = tmp_2;
	} else if (t_4 >= t_0) {
		tmp_1 = t_7;
	} else {
		tmp_1 = t_6;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dY_46_u) ^ Float32(2.0)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = t_3 ^ Float32(2.0)
	t_5 = sqrt(fmax(Float32(t_4 + (Float32(floor(w) * dX_46_u) ^ Float32(2.0))), Float32(t_2 + t_0)))
	t_6 = Float32(t_1 / t_5)
	t_7 = Float32(t_3 / t_5)
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(8000000.0))
		tmp_2 = Float32(0.0)
		if (t_4 >= t_2)
			tmp_2 = t_7;
		else
			tmp_2 = t_6;
		end
		tmp_1 = tmp_2;
	elseif (t_4 >= t_0)
		tmp_1 = t_7;
	else
		tmp_1 = t_6;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = (floor(w) * dY_46_u) ^ single(2.0);
	t_1 = floor(h) * dY_46_v;
	t_2 = t_1 ^ single(2.0);
	t_3 = floor(h) * dX_46_v;
	t_4 = t_3 ^ single(2.0);
	t_5 = sqrt(max((t_4 + ((floor(w) * dX_46_u) ^ single(2.0))), (t_2 + t_0)));
	t_6 = t_1 / t_5;
	t_7 = t_3 / t_5;
	tmp_2 = single(0.0);
	if (dY_46_u <= single(8000000.0))
		tmp_3 = single(0.0);
		if (t_4 >= t_2)
			tmp_3 = t_7;
		else
			tmp_3 = t_6;
		end
		tmp_2 = tmp_3;
	elseif (t_4 >= t_0)
		tmp_2 = t_7;
	else
		tmp_2 = t_6;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 8e6

   1. Initial program 78.0%

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

   3. Applied rewrites 78.3%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 66.7%

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

   7. Taylor expanded in dY.u around 0

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

   9. Applied rewrites 63.3%

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

   if 8e6 < dY.u

   1. Initial program 66.2%

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

   3. Applied rewrites 66.4%

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

   4. Taylor expanded in dX.u around 0

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in dY.u around inf

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

   9. Applied rewrites 59.2%

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

Alternative 11: 60.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

4. Taylor expanded in dX.u around 0

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

6. Applied rewrites 65.7%

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

7. Taylor expanded in dY.u around 0

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

9. Applied rewrites 60.5%

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

Alternative 12: 60.3% accurate, 1.3× speedup

Math

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

FPCore

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

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

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

4. Taylor expanded in dX.u around 0

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

6. Applied rewrites 65.7%

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

7. Taylor expanded in dY.u around 0

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

9. Applied rewrites 60.5%

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

11. Applied rewrites 60.3%

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

Alternative 13: 60.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 76.1%

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

3. Applied rewrites 76.4%

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

4. Taylor expanded in dX.u around 0

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

6. Applied rewrites 65.7%

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

7. Taylor expanded in dY.u around 0

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

9. Applied rewrites 60.5%

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

11. Applied rewrites 60.3%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, v)"
  :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 h) dX.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 h) dY.v))))