Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.1% → 76.4%

Time: 41.7s

Alternatives: 7

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

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

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(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(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

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

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(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(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

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

FPCore

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

Derivation

1. Initial program 77.3%

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

3. Simplified 77.2%

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

5. Taylor expanded in w around 0 77.2%

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

7. Simplified 77.2%

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

8. Taylor expanded in w around 0 77.2%

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

10. Simplified 77.3%

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

11. Taylor expanded in w around 0 77.3%

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

13. Simplified 77.4%

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

14. Final simplification 77.4%

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

Alternative 2: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := dX.v \cdot \left\lfloorh\right\rfloor\\ t_4 := \sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_3, t\_0\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\right)}\\ t_5 := \frac{t\_3}{t\_4}\\ t_6 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_7 := \frac{t\_1}{t\_4}\\ \mathbf{if}\;dX.u \leq 600000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_3}^{2} \geq t\_6:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array}\\ \mathbf{elif}\;{t\_0}^{2} \geq t\_6:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (* dX.v (floor h)))
        (t_4 (sqrt (fmax (pow (hypot t_3 t_0) 2.0) (pow (hypot t_2 t_1) 2.0))))
        (t_5 (/ t_3 t_4))
        (t_6 (pow (hypot t_1 t_2) 2.0))
        (t_7 (/ t_1 t_4)))
   (if (<= dX.u 600000.0)
     (if (>= (pow t_3 2.0) t_6) t_5 t_7)
     (if (>= (pow t_0 2.0) t_6) t_5 t_7))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = dX_46_v * floorf(h);
	float t_4 = sqrtf(fmaxf(powf(hypotf(t_3, t_0), 2.0f), powf(hypotf(t_2, t_1), 2.0f)));
	float t_5 = t_3 / t_4;
	float t_6 = powf(hypotf(t_1, t_2), 2.0f);
	float t_7 = t_1 / t_4;
	float tmp_1;
	if (dX_46_u <= 600000.0f) {
		float tmp_2;
		if (powf(t_3, 2.0f) >= t_6) {
			tmp_2 = t_5;
		} else {
			tmp_2 = t_7;
		}
		tmp_1 = tmp_2;
	} else if (powf(t_0, 2.0f) >= t_6) {
		tmp_1 = t_5;
	} 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(dX_46_u * floor(w))
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = Float32(dX_46_v * floor(h))
	t_4 = sqrt((((hypot(t_3, t_0) ^ Float32(2.0)) != (hypot(t_3, t_0) ^ Float32(2.0))) ? (hypot(t_2, t_1) ^ Float32(2.0)) : (((hypot(t_2, t_1) ^ Float32(2.0)) != (hypot(t_2, t_1) ^ Float32(2.0))) ? (hypot(t_3, t_0) ^ Float32(2.0)) : max((hypot(t_3, t_0) ^ Float32(2.0)), (hypot(t_2, t_1) ^ Float32(2.0))))))
	t_5 = Float32(t_3 / t_4)
	t_6 = hypot(t_1, t_2) ^ Float32(2.0)
	t_7 = Float32(t_1 / t_4)
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(600000.0))
		tmp_2 = Float32(0.0)
		if ((t_3 ^ Float32(2.0)) >= t_6)
			tmp_2 = t_5;
		else
			tmp_2 = t_7;
		end
		tmp_1 = tmp_2;
	elseif ((t_0 ^ Float32(2.0)) >= t_6)
		tmp_1 = t_5;
	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 = dX_46_u * floor(w);
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(w) * dY_46_u;
	t_3 = dX_46_v * floor(h);
	t_4 = sqrt(max((hypot(t_3, t_0) ^ single(2.0)), (hypot(t_2, t_1) ^ single(2.0))));
	t_5 = t_3 / t_4;
	t_6 = hypot(t_1, t_2) ^ single(2.0);
	t_7 = t_1 / t_4;
	tmp_2 = single(0.0);
	if (dX_46_u <= single(600000.0))
		tmp_3 = single(0.0);
		if ((t_3 ^ single(2.0)) >= t_6)
			tmp_3 = t_5;
		else
			tmp_3 = t_7;
		end
		tmp_2 = tmp_3;
	elseif ((t_0 ^ single(2.0)) >= t_6)
		tmp_2 = t_5;
	else
		tmp_2 = t_7;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 6e5

   1. Initial program 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in w around 0 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in w around 0 79.1%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in w around 0 79.2%

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

   13. Simplified 79.3%

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

   14. Taylor expanded in dX.u around 0 71.8%

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

       1. unpow2 71.8%

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

       2. unpow2 71.8%

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

       3. swap-sqr 71.8%

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

       4. unpow2 71.8%

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

   16. Simplified 71.8%

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

   if 6e5 < dX.u

   1. Initial program 70.0%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in w around 0 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in w around 0 69.8%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in w around 0 69.9%

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

   13. Simplified 70.1%

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

   14. Taylor expanded in dX.u around inf 68.6%

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

       1. unpow2 68.6%

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

       2. unpow2 68.6%

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

       3. swap-sqr 68.6%

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

       4. unpow2 68.6%

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

   16. Simplified 68.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 600000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := dX.v \cdot \left\lfloorh\right\rfloor\\ t_3 := {\left(\mathsf{hypot}\left(t\_2, t\_0\right)\right)}^{2}\\ t_4 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_5 := {\left(\mathsf{hypot}\left(t\_1, t\_4\right)\right)}^{2}\\ t_6 := \sqrt{\mathsf{max}\left(t\_3, t\_5\right)}\\ t_7 := \sqrt{\mathsf{max}\left(t\_3, {\left(\mathsf{hypot}\left(t\_4, t\_1\right)\right)}^{2}\right)}\\ \mathbf{if}\;dX.u \leq 700000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_2}^{2} \geq t\_5:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_6}\\ \end{array}\\ \mathbf{elif}\;{t\_0}^{2} \geq t\_5:\\ \;\;\;\;\frac{t\_2}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_7}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor h) dY.v))
        (t_2 (* dX.v (floor h)))
        (t_3 (pow (hypot t_2 t_0) 2.0))
        (t_4 (* (floor w) dY.u))
        (t_5 (pow (hypot t_1 t_4) 2.0))
        (t_6 (sqrt (fmax t_3 t_5)))
        (t_7 (sqrt (fmax t_3 (pow (hypot t_4 t_1) 2.0)))))
   (if (<= dX.u 700000.0)
     (if (>= (pow t_2 2.0) t_5) (* dX.v (/ (floor h) t_6)) (/ t_1 t_6))
     (if (>= (pow t_0 2.0) t_5) (/ t_2 t_7) (/ t_1 t_7)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = dX_46_v * floorf(h);
	float t_3 = powf(hypotf(t_2, t_0), 2.0f);
	float t_4 = floorf(w) * dY_46_u;
	float t_5 = powf(hypotf(t_1, t_4), 2.0f);
	float t_6 = sqrtf(fmaxf(t_3, t_5));
	float t_7 = sqrtf(fmaxf(t_3, powf(hypotf(t_4, t_1), 2.0f)));
	float tmp_1;
	if (dX_46_u <= 700000.0f) {
		float tmp_2;
		if (powf(t_2, 2.0f) >= t_5) {
			tmp_2 = dX_46_v * (floorf(h) / t_6);
		} else {
			tmp_2 = t_1 / t_6;
		}
		tmp_1 = tmp_2;
	} else if (powf(t_0, 2.0f) >= t_5) {
		tmp_1 = t_2 / t_7;
	} else {
		tmp_1 = t_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(dX_46_u * floor(w))
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = hypot(t_2, t_0) ^ Float32(2.0)
	t_4 = Float32(floor(w) * dY_46_u)
	t_5 = hypot(t_1, t_4) ^ Float32(2.0)
	t_6 = sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5))))
	t_7 = sqrt(((t_3 != t_3) ? (hypot(t_4, t_1) ^ Float32(2.0)) : (((hypot(t_4, t_1) ^ Float32(2.0)) != (hypot(t_4, t_1) ^ Float32(2.0))) ? t_3 : max(t_3, (hypot(t_4, t_1) ^ Float32(2.0))))))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(700000.0))
		tmp_2 = Float32(0.0)
		if ((t_2 ^ Float32(2.0)) >= t_5)
			tmp_2 = Float32(dX_46_v * Float32(floor(h) / t_6));
		else
			tmp_2 = Float32(t_1 / t_6);
		end
		tmp_1 = tmp_2;
	elseif ((t_0 ^ Float32(2.0)) >= t_5)
		tmp_1 = Float32(t_2 / t_7);
	else
		tmp_1 = Float32(t_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 = dX_46_u * floor(w);
	t_1 = floor(h) * dY_46_v;
	t_2 = dX_46_v * floor(h);
	t_3 = hypot(t_2, t_0) ^ single(2.0);
	t_4 = floor(w) * dY_46_u;
	t_5 = hypot(t_1, t_4) ^ single(2.0);
	t_6 = sqrt(max(t_3, t_5));
	t_7 = sqrt(max(t_3, (hypot(t_4, t_1) ^ single(2.0))));
	tmp_2 = single(0.0);
	if (dX_46_u <= single(700000.0))
		tmp_3 = single(0.0);
		if ((t_2 ^ single(2.0)) >= t_5)
			tmp_3 = dX_46_v * (floor(h) / t_6);
		else
			tmp_3 = t_1 / t_6;
		end
		tmp_2 = tmp_3;
	elseif ((t_0 ^ single(2.0)) >= t_5)
		tmp_2 = t_2 / t_7;
	else
		tmp_2 = t_1 / t_7;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 7e5

   1. Initial program 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in w around 0 79.0%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in dX.u around 0 71.2%

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

      1. unpow2 71.8%

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

      2. unpow2 71.8%

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

      3. swap-sqr 71.8%

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

      4. unpow2 71.8%

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

   10. Simplified 71.2%

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

   11. Taylor expanded in dX.v around 0 71.3%

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

   13. Simplified 71.7%

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

   if 7e5 < dX.u

   1. Initial program 70.0%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in w around 0 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in w around 0 69.8%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in w around 0 69.9%

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

   13. Simplified 70.1%

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

   14. Taylor expanded in dX.u around inf 68.6%

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

       1. unpow2 68.6%

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

       2. unpow2 68.6%

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

       3. swap-sqr 68.6%

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

       4. unpow2 68.6%

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

   16. Simplified 68.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 700000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_4 := dX.v \cdot \left\lfloorh\right\rfloor\\ t_5 := {t\_4}^{2}\\ t_6 := {\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}\\ t_7 := \sqrt{\mathsf{max}\left(t\_6, {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\right)}\\ \mathbf{if}\;dX.u \leq 700000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_3:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left(t\_6, t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array}\\ \mathbf{elif}\;{t\_0}^{2} \geq t\_3:\\ \;\;\;\;\frac{t\_4}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_7}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow t_4 2.0))
        (t_6 (pow (hypot t_4 t_0) 2.0))
        (t_7 (sqrt (fmax t_6 (pow (hypot t_2 t_1) 2.0)))))
   (if (<= dX.u 700000.0)
     (if (>= t_5 t_3)
       (* dX.v (/ (floor h) (sqrt (fmax t_6 t_3))))
       (/ t_1 (sqrt (fmax t_5 t_3))))
     (if (>= (pow t_0 2.0) t_3) (/ t_4 t_7) (/ t_1 t_7)))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(hypotf(t_1, t_2), 2.0f);
	float t_4 = dX_46_v * floorf(h);
	float t_5 = powf(t_4, 2.0f);
	float t_6 = powf(hypotf(t_4, t_0), 2.0f);
	float t_7 = sqrtf(fmaxf(t_6, powf(hypotf(t_2, t_1), 2.0f)));
	float tmp_1;
	if (dX_46_u <= 700000.0f) {
		float tmp_2;
		if (t_5 >= t_3) {
			tmp_2 = dX_46_v * (floorf(h) / sqrtf(fmaxf(t_6, t_3)));
		} else {
			tmp_2 = t_1 / sqrtf(fmaxf(t_5, t_3));
		}
		tmp_1 = tmp_2;
	} else if (powf(t_0, 2.0f) >= t_3) {
		tmp_1 = t_4 / t_7;
	} else {
		tmp_1 = t_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(dX_46_u * floor(w))
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = t_4 ^ Float32(2.0)
	t_6 = hypot(t_4, t_0) ^ Float32(2.0)
	t_7 = sqrt(((t_6 != t_6) ? (hypot(t_2, t_1) ^ Float32(2.0)) : (((hypot(t_2, t_1) ^ Float32(2.0)) != (hypot(t_2, t_1) ^ Float32(2.0))) ? t_6 : max(t_6, (hypot(t_2, t_1) ^ Float32(2.0))))))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(700000.0))
		tmp_2 = Float32(0.0)
		if (t_5 >= t_3)
			tmp_2 = Float32(dX_46_v * Float32(floor(h) / sqrt(((t_6 != t_6) ? t_3 : ((t_3 != t_3) ? t_6 : max(t_6, t_3))))));
		else
			tmp_2 = Float32(t_1 / sqrt(((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3)))));
		end
		tmp_1 = tmp_2;
	elseif ((t_0 ^ Float32(2.0)) >= t_3)
		tmp_1 = Float32(t_4 / t_7);
	else
		tmp_1 = Float32(t_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 = dX_46_u * floor(w);
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(w) * dY_46_u;
	t_3 = hypot(t_1, t_2) ^ single(2.0);
	t_4 = dX_46_v * floor(h);
	t_5 = t_4 ^ single(2.0);
	t_6 = hypot(t_4, t_0) ^ single(2.0);
	t_7 = sqrt(max(t_6, (hypot(t_2, t_1) ^ single(2.0))));
	tmp_2 = single(0.0);
	if (dX_46_u <= single(700000.0))
		tmp_3 = single(0.0);
		if (t_5 >= t_3)
			tmp_3 = dX_46_v * (floor(h) / sqrt(max(t_6, t_3)));
		else
			tmp_3 = t_1 / sqrt(max(t_5, t_3));
		end
		tmp_2 = tmp_3;
	elseif ((t_0 ^ single(2.0)) >= t_3)
		tmp_2 = t_4 / t_7;
	else
		tmp_2 = t_1 / t_7;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 7e5

   1. Initial program 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in w around 0 79.0%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in dX.u around 0 71.2%

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

      1. unpow2 71.8%

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

      2. unpow2 71.8%

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

      3. swap-sqr 71.8%

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

      4. unpow2 71.8%

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

   10. Simplified 71.2%

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

   11. Taylor expanded in dX.v around 0 71.3%

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

   13. Simplified 71.7%

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

   14. Taylor expanded in dX.v around inf 71.6%

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

       1. unpow2 71.6%

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

       2. unpow2 71.6%

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

       3. swap-sqr 71.6%

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

       4. unpow2 71.6%

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

   16. Simplified 71.6%

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

   if 7e5 < dX.u

   1. Initial program 70.0%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in w around 0 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in w around 0 69.8%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in w around 0 69.9%

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

   13. Simplified 70.1%

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

   14. Taylor expanded in dX.u around inf 68.6%

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

       1. unpow2 68.6%

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

       2. unpow2 68.6%

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

       3. swap-sqr 68.6%

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

       4. unpow2 68.6%

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

   16. Simplified 68.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 700000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_4 := dX.v \cdot \left\lfloorh\right\rfloor\\ t_5 := {\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}\\ t_6 := \sqrt{\mathsf{max}\left(t\_5, {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\right)}\\ t_7 := {t\_4}^{2}\\ \mathbf{if}\;dX.u \leq 1.5:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_7 \geq t\_3:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left(t\_7, t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array}\\ \mathbf{elif}\;{t\_0}^{2} \geq t\_3:\\ \;\;\;\;\frac{t\_4}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_6}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow (hypot t_4 t_0) 2.0))
        (t_6 (sqrt (fmax t_5 (pow (hypot t_2 t_1) 2.0))))
        (t_7 (pow t_4 2.0)))
   (if (<= dX.u 1.5)
     (if (>= t_7 t_3)
       (* dX.v (/ (floor h) (sqrt (fmax t_7 t_3))))
       (/ t_1 (sqrt (fmax t_5 t_3))))
     (if (>= (pow t_0 2.0) t_3) (/ t_4 t_6) (/ 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 = dX_46_u * floorf(w);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(hypotf(t_1, t_2), 2.0f);
	float t_4 = dX_46_v * floorf(h);
	float t_5 = powf(hypotf(t_4, t_0), 2.0f);
	float t_6 = sqrtf(fmaxf(t_5, powf(hypotf(t_2, t_1), 2.0f)));
	float t_7 = powf(t_4, 2.0f);
	float tmp_1;
	if (dX_46_u <= 1.5f) {
		float tmp_2;
		if (t_7 >= t_3) {
			tmp_2 = dX_46_v * (floorf(h) / sqrtf(fmaxf(t_7, t_3)));
		} else {
			tmp_2 = t_1 / sqrtf(fmaxf(t_5, t_3));
		}
		tmp_1 = tmp_2;
	} else if (powf(t_0, 2.0f) >= t_3) {
		tmp_1 = t_4 / t_6;
	} 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(dX_46_u * floor(w))
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = hypot(t_4, t_0) ^ Float32(2.0)
	t_6 = sqrt(((t_5 != t_5) ? (hypot(t_2, t_1) ^ Float32(2.0)) : (((hypot(t_2, t_1) ^ Float32(2.0)) != (hypot(t_2, t_1) ^ Float32(2.0))) ? t_5 : max(t_5, (hypot(t_2, t_1) ^ Float32(2.0))))))
	t_7 = t_4 ^ Float32(2.0)
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(1.5))
		tmp_2 = Float32(0.0)
		if (t_7 >= t_3)
			tmp_2 = Float32(dX_46_v * Float32(floor(h) / sqrt(((t_7 != t_7) ? t_3 : ((t_3 != t_3) ? t_7 : max(t_7, t_3))))));
		else
			tmp_2 = Float32(t_1 / sqrt(((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3)))));
		end
		tmp_1 = tmp_2;
	elseif ((t_0 ^ Float32(2.0)) >= t_3)
		tmp_1 = Float32(t_4 / t_6);
	else
		tmp_1 = Float32(t_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 = dX_46_u * floor(w);
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(w) * dY_46_u;
	t_3 = hypot(t_1, t_2) ^ single(2.0);
	t_4 = dX_46_v * floor(h);
	t_5 = hypot(t_4, t_0) ^ single(2.0);
	t_6 = sqrt(max(t_5, (hypot(t_2, t_1) ^ single(2.0))));
	t_7 = t_4 ^ single(2.0);
	tmp_2 = single(0.0);
	if (dX_46_u <= single(1.5))
		tmp_3 = single(0.0);
		if (t_7 >= t_3)
			tmp_3 = dX_46_v * (floor(h) / sqrt(max(t_7, t_3)));
		else
			tmp_3 = t_1 / sqrt(max(t_5, t_3));
		end
		tmp_2 = tmp_3;
	elseif ((t_0 ^ single(2.0)) >= t_3)
		tmp_2 = t_4 / t_6;
	else
		tmp_2 = t_1 / t_6;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1.5

   1. Initial program 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in w around 0 79.0%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in dX.u around 0 70.5%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. swap-sqr 71.1%

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

      4. unpow2 71.1%

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

   10. Simplified 70.5%

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

   11. Taylor expanded in dX.v around 0 70.6%

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

   13. Simplified 71.0%

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

   14. Taylor expanded in dX.v around inf 66.0%

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

       1. unpow2 70.9%

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

       2. unpow2 70.9%

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

       3. swap-sqr 70.9%

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

       4. unpow2 70.9%

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

   16. Simplified 66.1%

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

   if 1.5 < dX.u

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in w around 0 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in w around 0 72.1%

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

   10. Simplified 72.1%

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

   11. Taylor expanded in w around 0 72.1%

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

   13. Simplified 72.3%

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

   14. Taylor expanded in dX.u around inf 69.8%

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

       1. unpow2 69.8%

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

       2. unpow2 69.8%

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

       3. swap-sqr 69.8%

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

       4. unpow2 69.8%

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

   16. Simplified 69.8%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 1.5:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.v \cdot \left\lfloorh\right\rfloor\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_4 := dX.u \cdot \left\lfloorw\right\rfloor\\ t_5 := {\left(\mathsf{hypot}\left(t\_0, t\_4\right)\right)}^{2}\\ t_6 := \sqrt{\mathsf{max}\left(t\_5, {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\right)}\\ \mathbf{if}\;dX.u \leq 1.5:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_0}^{2} \geq t\_3:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ \end{array}\\ \mathbf{elif}\;{t\_4}^{2} \geq t\_3:\\ \;\;\;\;\frac{t\_0}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_6}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.v (floor h)))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.u (floor w)))
        (t_5 (pow (hypot t_0 t_4) 2.0))
        (t_6 (sqrt (fmax t_5 (pow (hypot t_2 t_1) 2.0)))))
   (if (<= dX.u 1.5)
     (if (>= (pow t_0 2.0) t_3)
       (*
        dX.v
        (/ (floor h) (sqrt (fmax (* (pow dX.v 2.0) (pow (floor h) 2.0)) t_3))))
       (/ t_1 (sqrt (fmax t_5 t_3))))
     (if (>= (pow t_4 2.0) t_3) (/ t_0 t_6) (/ 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 = dX_46_v * floorf(h);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(hypotf(t_1, t_2), 2.0f);
	float t_4 = dX_46_u * floorf(w);
	float t_5 = powf(hypotf(t_0, t_4), 2.0f);
	float t_6 = sqrtf(fmaxf(t_5, powf(hypotf(t_2, t_1), 2.0f)));
	float tmp_1;
	if (dX_46_u <= 1.5f) {
		float tmp_2;
		if (powf(t_0, 2.0f) >= t_3) {
			tmp_2 = dX_46_v * (floorf(h) / sqrtf(fmaxf((powf(dX_46_v, 2.0f) * powf(floorf(h), 2.0f)), t_3)));
		} else {
			tmp_2 = t_1 / sqrtf(fmaxf(t_5, t_3));
		}
		tmp_1 = tmp_2;
	} else if (powf(t_4, 2.0f) >= t_3) {
		tmp_1 = t_0 / t_6;
	} 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(dX_46_v * floor(h))
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = hypot(t_0, t_4) ^ Float32(2.0)
	t_6 = sqrt(((t_5 != t_5) ? (hypot(t_2, t_1) ^ Float32(2.0)) : (((hypot(t_2, t_1) ^ Float32(2.0)) != (hypot(t_2, t_1) ^ Float32(2.0))) ? t_5 : max(t_5, (hypot(t_2, t_1) ^ Float32(2.0))))))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(1.5))
		tmp_2 = Float32(0.0)
		if ((t_0 ^ Float32(2.0)) >= t_3)
			tmp_2 = Float32(dX_46_v * Float32(floor(h) / sqrt(((Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0))) != Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0)))) ? t_3 : ((t_3 != t_3) ? Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0))) : max(Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0))), t_3))))));
		else
			tmp_2 = Float32(t_1 / sqrt(((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3)))));
		end
		tmp_1 = tmp_2;
	elseif ((t_4 ^ Float32(2.0)) >= t_3)
		tmp_1 = Float32(t_0 / t_6);
	else
		tmp_1 = Float32(t_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 = dX_46_v * floor(h);
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(w) * dY_46_u;
	t_3 = hypot(t_1, t_2) ^ single(2.0);
	t_4 = dX_46_u * floor(w);
	t_5 = hypot(t_0, t_4) ^ single(2.0);
	t_6 = sqrt(max(t_5, (hypot(t_2, t_1) ^ single(2.0))));
	tmp_2 = single(0.0);
	if (dX_46_u <= single(1.5))
		tmp_3 = single(0.0);
		if ((t_0 ^ single(2.0)) >= t_3)
			tmp_3 = dX_46_v * (floor(h) / sqrt(max(((dX_46_v ^ single(2.0)) * (floor(h) ^ single(2.0))), t_3)));
		else
			tmp_3 = t_1 / sqrt(max(t_5, t_3));
		end
		tmp_2 = tmp_3;
	elseif ((t_4 ^ single(2.0)) >= t_3)
		tmp_2 = t_0 / t_6;
	else
		tmp_2 = t_1 / t_6;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1.5

   1. Initial program 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in w around 0 79.0%

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

   7. Simplified 78.8%

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

   8. Taylor expanded in dX.u around 0 70.5%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. swap-sqr 71.1%

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

      4. unpow2 71.1%

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

   10. Simplified 70.5%

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

   11. Taylor expanded in dX.v around 0 70.6%

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

   13. Simplified 71.0%

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

   14. Taylor expanded in dX.v around inf 66.0%

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

   if 1.5 < dX.u

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in w around 0 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in w around 0 72.1%

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

   10. Simplified 72.1%

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

   11. Taylor expanded in w around 0 72.1%

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

   13. Simplified 72.3%

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

   14. Taylor expanded in dX.u around inf 69.8%

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

       1. unpow2 69.8%

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

       2. unpow2 69.8%

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

       3. swap-sqr 69.8%

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

       4. unpow2 69.8%

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

   16. Simplified 69.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 1.5:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloorh\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{dX.v \cdot \left\lfloorh\right\rfloor}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloorh\right\rfloor, dX.u \cdot \left\lfloorw\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 77.3%

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

3. Simplified 77.2%

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

5. Taylor expanded in w around 0 77.2%

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

7. Simplified 77.2%

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

8. Taylor expanded in w around 0 77.2%

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

10. Simplified 77.3%

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

11. Taylor expanded in w around 0 77.3%

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

13. Simplified 77.4%

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

14. Taylor expanded in dX.u around inf 64.8%

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

    1. unpow2 64.8%

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

    2. unpow2 64.8%

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

    3. swap-sqr 64.8%

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

    4. unpow2 64.8%

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

16. Simplified 64.8%

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

17. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2024129 
(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))))