Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.1% → 76.3%

Time: 42.8s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((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

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((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: 76.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = dX_46_u * floorf(w);
	float t_3 = dX_46_v * floorf(h);
	float t_4 = powf(hypotf(t_2, t_3), 2.0f);
	float tmp;
	if (t_4 >= powf(hypotf(t_0, t_1), 2.0f)) {
		tmp = t_3 / sqrtf(fmaxf(fmaf(t_2, t_2, (t_3 * t_3)), fmaf(t_0, t_0, (floorf(h) * (dY_46_v * t_1)))));
	} else {
		tmp = powf((sqrtf(fmaxf(t_4, powf(hypotf(t_1, t_0), 2.0f))) / t_1), -1.0f);
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(dX_46_u * floor(w))
	t_3 = Float32(dX_46_v * floor(h))
	t_4 = hypot(t_2, t_3) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_4 >= (hypot(t_0, t_1) ^ Float32(2.0)))
		tmp = Float32(t_3 / sqrt(((fma(t_2, t_2, Float32(t_3 * t_3)) != fma(t_2, t_2, Float32(t_3 * t_3))) ? fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) : ((fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) != fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1)))) ? fma(t_2, t_2, Float32(t_3 * t_3)) : max(fma(t_2, t_2, Float32(t_3 * t_3)), fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))))))));
	else
		tmp = Float32(sqrt(((t_4 != t_4) ? (hypot(t_1, t_0) ^ Float32(2.0)) : (((hypot(t_1, t_0) ^ Float32(2.0)) != (hypot(t_1, t_0) ^ Float32(2.0))) ? t_4 : max(t_4, (hypot(t_1, t_0) ^ Float32(2.0)))))) / t_1) ^ Float32(-1.0);
	end
	return tmp
end

Derivation

1. Initial program 78.5%

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

3. Simplified 78.8%

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

6. Applied egg-rr 78.9%

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

7. Taylor expanded in w around 0 78.9%

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

9. Simplified 78.9%

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

10. Final simplification 78.9%

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

Alternative 2: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = dX_46_v * floorf(h);
	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 tmp;
	if (t_3 >= t_5) {
		tmp = t_2 / sqrtf(fmaxf(powf(hypotf(t_0, t_2), 2.0f), powf(hypotf(t_1, t_4), 2.0f)));
	} else {
		tmp = t_4 * (1.0f / sqrtf(fmaxf(t_3, t_5)));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(dX_46_v * floor(h))
	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))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_2 / sqrt((((hypot(t_0, t_2) ^ Float32(2.0)) != (hypot(t_0, t_2) ^ Float32(2.0))) ? (hypot(t_1, t_4) ^ Float32(2.0)) : (((hypot(t_1, t_4) ^ Float32(2.0)) != (hypot(t_1, t_4) ^ Float32(2.0))) ? (hypot(t_0, t_2) ^ Float32(2.0)) : max((hypot(t_0, t_2) ^ Float32(2.0)), (hypot(t_1, t_4) ^ Float32(2.0)))))));
	else
		tmp = Float32(t_4 * Float32(Float32(1.0) / sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = dX_46_u * floor(w);
	t_1 = floor(w) * dY_46_u;
	t_2 = dX_46_v * floor(h);
	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);
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_2 / sqrt(max((hypot(t_0, t_2) ^ single(2.0)), (hypot(t_1, t_4) ^ single(2.0))));
	else
		tmp = t_4 * (single(1.0) / sqrt(max(t_3, t_5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in w around 0 78.5%

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

   1. associate-*l/ 78.8%

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

   2. *-un-lft-identity 78.8%

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

   3. pow-prod-down 78.8%

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

   4. *-commutative 78.8%

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

   5. pow2 78.8%

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

5. Applied egg-rr 78.8%

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

6. Final simplification 78.8%

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

Alternative 3: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = dX_46_v * floorf(h);
	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 tmp;
	if (t_3 >= t_5) {
		tmp = 1.0f / (sqrtf(fmaxf(powf(hypotf(t_0, t_2), 2.0f), powf(hypotf(t_1, t_4), 2.0f))) / t_2);
	} else {
		tmp = t_4 * (1.0f / sqrtf(fmaxf(t_3, t_5)));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_u * floor(w))
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(dX_46_v * floor(h))
	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))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(Float32(1.0) / Float32(sqrt((((hypot(t_0, t_2) ^ Float32(2.0)) != (hypot(t_0, t_2) ^ Float32(2.0))) ? (hypot(t_1, t_4) ^ Float32(2.0)) : (((hypot(t_1, t_4) ^ Float32(2.0)) != (hypot(t_1, t_4) ^ Float32(2.0))) ? (hypot(t_0, t_2) ^ Float32(2.0)) : max((hypot(t_0, t_2) ^ Float32(2.0)), (hypot(t_1, t_4) ^ Float32(2.0)))))) / t_2));
	else
		tmp = Float32(t_4 * Float32(Float32(1.0) / sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = dX_46_u * floor(w);
	t_1 = floor(w) * dY_46_u;
	t_2 = dX_46_v * floor(h);
	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);
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = single(1.0) / (sqrt(max((hypot(t_0, t_2) ^ single(2.0)), (hypot(t_1, t_4) ^ single(2.0)))) / t_2);
	else
		tmp = t_4 * (single(1.0) / sqrt(max(t_3, t_5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in w around 0 78.5%

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

5. Applied egg-rr 78.7%

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

6. Final simplification 78.7%

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

Alternative 4: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.5%

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

3. Simplified 78.8%

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

6. Applied egg-rr 78.9%

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

7. Taylor expanded in w around 0 78.9%

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

9. Simplified 78.9%

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

10. Taylor expanded in dX.u around 0 78.5%

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

12. Simplified 78.7%

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

Alternative 5: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_4 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_5 := {\left(\mathsf{hypot}\left(t\_0, t\_4\right)\right)}^{2}\\ t_6 := \sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}, t\_3\right)}\\ \mathbf{if}\;dX.u \leq 1500:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_4}^{2} \geq t\_3:\\ \;\;\;\;\frac{1}{\frac{t\_6}{t\_4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_6}\\ \end{array}\\ \mathbf{elif}\;t\_5 \geq {t\_1}^{2}:\\ \;\;\;\;\frac{t\_4}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \frac{1}{\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow (hypot t_0 t_4) 2.0))
        (t_6 (sqrt (fmax (pow (hypot t_4 t_0) 2.0) t_3))))
   (if (<= dX.u 1500.0)
     (if (>= (pow t_4 2.0) t_3) (/ 1.0 (/ t_6 t_4)) (/ t_2 t_6))
     (if (>= t_5 (pow t_1 2.0))
       (/ t_4 t_6)
       (* (floor h) (* dY.v (/ 1.0 (sqrt (fmax t_5 t_3)))))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_u * floorf(w);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	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_0, t_4), 2.0f);
	float t_6 = sqrtf(fmaxf(powf(hypotf(t_4, t_0), 2.0f), t_3));
	float tmp_1;
	if (dX_46_u <= 1500.0f) {
		float tmp_2;
		if (powf(t_4, 2.0f) >= t_3) {
			tmp_2 = 1.0f / (t_6 / t_4);
		} else {
			tmp_2 = t_2 / t_6;
		}
		tmp_1 = tmp_2;
	} else if (t_5 >= powf(t_1, 2.0f)) {
		tmp_1 = t_4 / t_6;
	} else {
		tmp_1 = floorf(h) * (dY_46_v * (1.0f / sqrtf(fmaxf(t_5, t_3))));
	}
	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(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = hypot(t_0, t_4) ^ Float32(2.0)
	t_6 = sqrt((((hypot(t_4, t_0) ^ Float32(2.0)) != (hypot(t_4, t_0) ^ Float32(2.0))) ? t_3 : ((t_3 != t_3) ? (hypot(t_4, t_0) ^ Float32(2.0)) : max((hypot(t_4, t_0) ^ Float32(2.0)), t_3))))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(1500.0))
		tmp_2 = Float32(0.0)
		if ((t_4 ^ Float32(2.0)) >= t_3)
			tmp_2 = Float32(Float32(1.0) / Float32(t_6 / t_4));
		else
			tmp_2 = Float32(t_2 / t_6);
		end
		tmp_1 = tmp_2;
	elseif (t_5 >= (t_1 ^ Float32(2.0)))
		tmp_1 = Float32(t_4 / t_6);
	else
		tmp_1 = Float32(floor(h) * Float32(dY_46_v * Float32(Float32(1.0) / sqrt(((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3)))))));
	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(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = hypot(t_1, t_2) ^ single(2.0);
	t_4 = dX_46_v * floor(h);
	t_5 = hypot(t_0, t_4) ^ single(2.0);
	t_6 = sqrt(max((hypot(t_4, t_0) ^ single(2.0)), t_3));
	tmp_2 = single(0.0);
	if (dX_46_u <= single(1500.0))
		tmp_3 = single(0.0);
		if ((t_4 ^ single(2.0)) >= t_3)
			tmp_3 = single(1.0) / (t_6 / t_4);
		else
			tmp_3 = t_2 / t_6;
		end
		tmp_2 = tmp_3;
	elseif (t_5 >= (t_1 ^ single(2.0)))
		tmp_2 = t_4 / t_6;
	else
		tmp_2 = floor(h) * (dY_46_v * (single(1.0) / sqrt(max(t_5, t_3))));
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1500

   1. Initial program 80.9%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in w around 0 81.0%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in dX.u around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. unpow2 74.9%

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

      3. unpow2 74.9%

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

      4. swap-sqr 74.9%

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

      5. unpow2 74.9%

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

      6. *-commutative 74.9%

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

   10. Simplified 74.9%

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

   11. Taylor expanded in dX.v around 0 75.1%

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

   13. Simplified 75.4%

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

   if 1500 < dX.u

   1. Initial program 68.7%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in w around 0 68.6%

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

   9. Simplified 68.5%

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

   11. Applied egg-rr 68.7%

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

   12. Taylor expanded in dY.u around inf 63.4%

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

       1. *-commutative 63.4%

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

       2. unpow2 63.4%

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

       3. unpow2 63.4%

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

       4. swap-sqr 63.4%

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

       5. unpow2 63.4%

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

   14. Simplified 63.4%

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

Alternative 6: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_4 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_5 := {\left(\mathsf{hypot}\left(t\_0, t\_4\right)\right)}^{2}\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t\_5, t\_3\right)}}\\ t_7 := \sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}, t\_3\right)}\\ \mathbf{if}\;dX.u \leq 1500:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_4}^{2} \geq t\_3:\\ \;\;\;\;\frac{1}{\frac{t\_7}{t\_4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_7}\\ \end{array}\\ \mathbf{elif}\;t\_5 \geq {t\_1}^{2}:\\ \;\;\;\;t\_4 \cdot t\_6\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot t\_6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow (hypot t_0 t_4) 2.0))
        (t_6 (/ 1.0 (sqrt (fmax t_5 t_3))))
        (t_7 (sqrt (fmax (pow (hypot t_4 t_0) 2.0) t_3))))
   (if (<= dX.u 1500.0)
     (if (>= (pow t_4 2.0) t_3) (/ 1.0 (/ t_7 t_4)) (/ t_2 t_7))
     (if (>= t_5 (pow t_1 2.0)) (* t_4 t_6) (* (floor h) (* dY.v 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(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	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_0, t_4), 2.0f);
	float t_6 = 1.0f / sqrtf(fmaxf(t_5, t_3));
	float t_7 = sqrtf(fmaxf(powf(hypotf(t_4, t_0), 2.0f), t_3));
	float tmp_1;
	if (dX_46_u <= 1500.0f) {
		float tmp_2;
		if (powf(t_4, 2.0f) >= t_3) {
			tmp_2 = 1.0f / (t_7 / t_4);
		} else {
			tmp_2 = t_2 / t_7;
		}
		tmp_1 = tmp_2;
	} else if (t_5 >= powf(t_1, 2.0f)) {
		tmp_1 = t_4 * t_6;
	} else {
		tmp_1 = floorf(h) * (dY_46_v * 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(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = hypot(t_0, t_4) ^ Float32(2.0)
	t_6 = Float32(Float32(1.0) / sqrt(((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3)))))
	t_7 = sqrt((((hypot(t_4, t_0) ^ Float32(2.0)) != (hypot(t_4, t_0) ^ Float32(2.0))) ? t_3 : ((t_3 != t_3) ? (hypot(t_4, t_0) ^ Float32(2.0)) : max((hypot(t_4, t_0) ^ Float32(2.0)), t_3))))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(1500.0))
		tmp_2 = Float32(0.0)
		if ((t_4 ^ Float32(2.0)) >= t_3)
			tmp_2 = Float32(Float32(1.0) / Float32(t_7 / t_4));
		else
			tmp_2 = Float32(t_2 / t_7);
		end
		tmp_1 = tmp_2;
	elseif (t_5 >= (t_1 ^ Float32(2.0)))
		tmp_1 = Float32(t_4 * t_6);
	else
		tmp_1 = Float32(floor(h) * Float32(dY_46_v * 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(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = hypot(t_1, t_2) ^ single(2.0);
	t_4 = dX_46_v * floor(h);
	t_5 = hypot(t_0, t_4) ^ single(2.0);
	t_6 = single(1.0) / sqrt(max(t_5, t_3));
	t_7 = sqrt(max((hypot(t_4, t_0) ^ single(2.0)), t_3));
	tmp_2 = single(0.0);
	if (dX_46_u <= single(1500.0))
		tmp_3 = single(0.0);
		if ((t_4 ^ single(2.0)) >= t_3)
			tmp_3 = single(1.0) / (t_7 / t_4);
		else
			tmp_3 = t_2 / t_7;
		end
		tmp_2 = tmp_3;
	elseif (t_5 >= (t_1 ^ single(2.0)))
		tmp_2 = t_4 * t_6;
	else
		tmp_2 = floor(h) * (dY_46_v * t_6);
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1500

   1. Initial program 80.9%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in w around 0 81.0%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in dX.u around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. unpow2 74.9%

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

      3. unpow2 74.9%

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

      4. swap-sqr 74.9%

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

      5. unpow2 74.9%

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

      6. *-commutative 74.9%

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

   10. Simplified 74.9%

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

   11. Taylor expanded in dX.v around 0 75.1%

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

   13. Simplified 75.4%

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

   if 1500 < dX.u

   1. Initial program 68.7%

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

   3. Simplified 68.8%

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

   6. Applied egg-rr 68.9%

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

   7. Taylor expanded in w around 0 68.6%

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

   9. Simplified 68.5%

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

   10. Taylor expanded in dY.u around inf 63.2%

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

       1. *-commutative 63.4%

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

       2. unpow2 63.4%

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

       3. unpow2 63.4%

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

       4. swap-sqr 63.4%

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

       5. unpow2 63.4%

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

   12. Simplified 63.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 1500:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.u \cdot \left\lfloor w\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloor h\right\rfloor \cdot dY.v}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.v \cdot \left\lfloor h\right\rfloor , dX.u \cdot \left\lfloor w\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array}\\ \mathbf{elif}\;{\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2} \geq {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}:\\ \;\;\;\;\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\\ t_4 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_5 := {\left(\mathsf{hypot}\left(t\_0, t\_4\right)\right)}^{2}\\ t_6 := \sqrt{\frac{1}{\mathsf{max}\left(t\_5, {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\right)}}\\ t_7 := \sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}, t\_3\right)}\\ \mathbf{if}\;dX.u \leq 2000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_4}^{2} \geq t\_3:\\ \;\;\;\;\frac{1}{\frac{t\_7}{t\_4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_7}\\ \end{array}\\ \mathbf{elif}\;t\_5 \geq {t\_1}^{2}:\\ \;\;\;\;dX.v \cdot \left(\left\lfloor h\right\rfloor \cdot t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor h\right\rfloor \cdot \left(dY.v \cdot t\_6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow (hypot t_0 t_4) 2.0))
        (t_6 (sqrt (/ 1.0 (fmax t_5 (pow (hypot t_2 t_1) 2.0)))))
        (t_7 (sqrt (fmax (pow (hypot t_4 t_0) 2.0) t_3))))
   (if (<= dX.u 2000.0)
     (if (>= (pow t_4 2.0) t_3) (/ 1.0 (/ t_7 t_4)) (/ t_2 t_7))
     (if (>= t_5 (pow t_1 2.0))
       (* dX.v (* (floor h) t_6))
       (* (floor h) (* dY.v 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(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	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_0, t_4), 2.0f);
	float t_6 = sqrtf((1.0f / fmaxf(t_5, powf(hypotf(t_2, t_1), 2.0f))));
	float t_7 = sqrtf(fmaxf(powf(hypotf(t_4, t_0), 2.0f), t_3));
	float tmp_1;
	if (dX_46_u <= 2000.0f) {
		float tmp_2;
		if (powf(t_4, 2.0f) >= t_3) {
			tmp_2 = 1.0f / (t_7 / t_4);
		} else {
			tmp_2 = t_2 / t_7;
		}
		tmp_1 = tmp_2;
	} else if (t_5 >= powf(t_1, 2.0f)) {
		tmp_1 = dX_46_v * (floorf(h) * t_6);
	} else {
		tmp_1 = floorf(h) * (dY_46_v * 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(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = hypot(t_0, t_4) ^ Float32(2.0)
	t_6 = sqrt(Float32(Float32(1.0) / ((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 = sqrt((((hypot(t_4, t_0) ^ Float32(2.0)) != (hypot(t_4, t_0) ^ Float32(2.0))) ? t_3 : ((t_3 != t_3) ? (hypot(t_4, t_0) ^ Float32(2.0)) : max((hypot(t_4, t_0) ^ Float32(2.0)), t_3))))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(2000.0))
		tmp_2 = Float32(0.0)
		if ((t_4 ^ Float32(2.0)) >= t_3)
			tmp_2 = Float32(Float32(1.0) / Float32(t_7 / t_4));
		else
			tmp_2 = Float32(t_2 / t_7);
		end
		tmp_1 = tmp_2;
	elseif (t_5 >= (t_1 ^ Float32(2.0)))
		tmp_1 = Float32(dX_46_v * Float32(floor(h) * t_6));
	else
		tmp_1 = Float32(floor(h) * Float32(dY_46_v * 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(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = hypot(t_1, t_2) ^ single(2.0);
	t_4 = dX_46_v * floor(h);
	t_5 = hypot(t_0, t_4) ^ single(2.0);
	t_6 = sqrt((single(1.0) / max(t_5, (hypot(t_2, t_1) ^ single(2.0)))));
	t_7 = sqrt(max((hypot(t_4, t_0) ^ single(2.0)), t_3));
	tmp_2 = single(0.0);
	if (dX_46_u <= single(2000.0))
		tmp_3 = single(0.0);
		if ((t_4 ^ single(2.0)) >= t_3)
			tmp_3 = single(1.0) / (t_7 / t_4);
		else
			tmp_3 = t_2 / t_7;
		end
		tmp_2 = tmp_3;
	elseif (t_5 >= (t_1 ^ single(2.0)))
		tmp_2 = dX_46_v * (floor(h) * t_6);
	else
		tmp_2 = floor(h) * (dY_46_v * t_6);
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 2e3

   1. Initial program 81.0%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in w around 0 81.1%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in dX.u around 0 75.0%

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

      1. *-commutative 75.0%

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

      2. unpow2 75.0%

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

      3. unpow2 75.0%

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

      4. swap-sqr 75.0%

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

      5. unpow2 75.0%

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

      6. *-commutative 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in dX.v around 0 75.2%

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

   13. Simplified 75.5%

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

   if 2e3 < dX.u

   1. Initial program 68.1%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in w around 0 68.0%

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

   7. Simplified 67.8%

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

   8. Taylor expanded in dY.v around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. unpow2 62.5%

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

      4. swap-sqr 62.5%

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

      5. unpow2 62.5%

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

   10. Simplified 62.5%

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

Alternative 8: 62.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_3 := {t\_2}^{2}\\ t_4 := {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\\ t_5 := \sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_2, dX.u \cdot \left\lfloor w\right\rfloor \right)\right)}^{2}, t\_4\right)}\\ t_6 := \frac{t\_1}{t\_5}\\ \mathbf{if}\;dX.u \leq 0.05000000074505806:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq t\_4:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloor h\right\rfloor }{\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, t\_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array}\\ \mathbf{elif}\;t\_3 \geq {t\_0}^{2}:\\ \;\;\;\;dX.v \cdot \frac{\left\lfloor h\right\rfloor }{t\_5}\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* dX.v (floor h)))
        (t_3 (pow t_2 2.0))
        (t_4 (pow (hypot t_0 t_1) 2.0))
        (t_5 (sqrt (fmax (pow (hypot t_2 (* dX.u (floor w))) 2.0) t_4)))
        (t_6 (/ t_1 t_5)))
   (if (<= dX.u 0.05000000074505806)
     (if (>= t_3 t_4)
       (*
        dX.v
        (/ (floor h) (sqrt (fmax (* (pow dX.v 2.0) (pow (floor h) 2.0)) t_4))))
       t_6)
     (if (>= t_3 (pow t_0 2.0)) (* dX.v (/ (floor h) t_5)) 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 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = dX_46_v * floorf(h);
	float t_3 = powf(t_2, 2.0f);
	float t_4 = powf(hypotf(t_0, t_1), 2.0f);
	float t_5 = sqrtf(fmaxf(powf(hypotf(t_2, (dX_46_u * floorf(w))), 2.0f), t_4));
	float t_6 = t_1 / t_5;
	float tmp_1;
	if (dX_46_u <= 0.05000000074505806f) {
		float tmp_2;
		if (t_3 >= t_4) {
			tmp_2 = dX_46_v * (floorf(h) / sqrtf(fmaxf((powf(dX_46_v, 2.0f) * powf(floorf(h), 2.0f)), t_4)));
		} else {
			tmp_2 = t_6;
		}
		tmp_1 = tmp_2;
	} else if (t_3 >= powf(t_0, 2.0f)) {
		tmp_1 = dX_46_v * (floorf(h) / t_5);
	} else {
		tmp_1 = t_6;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = t_2 ^ Float32(2.0)
	t_4 = hypot(t_0, t_1) ^ Float32(2.0)
	t_5 = sqrt((((hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) != (hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0))) ? t_4 : ((t_4 != t_4) ? (hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) : max((hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)), t_4))))
	t_6 = Float32(t_1 / t_5)
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(0.05000000074505806))
		tmp_2 = Float32(0.0)
		if (t_3 >= t_4)
			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_4 : ((t_4 != t_4) ? 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_4))))));
		else
			tmp_2 = t_6;
		end
		tmp_1 = tmp_2;
	elseif (t_3 >= (t_0 ^ Float32(2.0)))
		tmp_1 = Float32(dX_46_v * Float32(floor(h) / t_5));
	else
		tmp_1 = t_6;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = dX_46_v * floor(h);
	t_3 = t_2 ^ single(2.0);
	t_4 = hypot(t_0, t_1) ^ single(2.0);
	t_5 = sqrt(max((hypot(t_2, (dX_46_u * floor(w))) ^ single(2.0)), t_4));
	t_6 = t_1 / t_5;
	tmp_2 = single(0.0);
	if (dX_46_u <= single(0.05000000074505806))
		tmp_3 = single(0.0);
		if (t_3 >= t_4)
			tmp_3 = dX_46_v * (floor(h) / sqrt(max(((dX_46_v ^ single(2.0)) * (floor(h) ^ single(2.0))), t_4)));
		else
			tmp_3 = t_6;
		end
		tmp_2 = tmp_3;
	elseif (t_3 >= (t_0 ^ single(2.0)))
		tmp_2 = dX_46_v * (floor(h) / t_5);
	else
		tmp_2 = t_6;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.0500000007

   1. Initial program 80.7%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in w around 0 80.8%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in dX.u around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. unpow2 74.8%

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

      3. unpow2 74.8%

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

      4. swap-sqr 74.8%

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

      5. unpow2 74.8%

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

      6. *-commutative 74.8%

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

   10. Simplified 74.8%

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

   11. Taylor expanded in dX.v around 0 75.0%

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

   13. Simplified 75.2%

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

   14. Taylor expanded in dX.v around inf 70.8%

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

   if 0.0500000007 < dX.u

   1. Initial program 71.6%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in w around 0 71.6%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in dX.u around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. unpow2 51.7%

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

      3. unpow2 51.7%

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

      4. swap-sqr 51.7%

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

      5. unpow2 51.7%

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

      6. *-commutative 51.7%

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

   10. Simplified 51.7%

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

   11. Taylor expanded in dX.v around 0 52.1%

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

   13. Simplified 52.2%

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

   14. Taylor expanded in dY.u around inf 53.5%

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

       1. *-commutative 65.7%

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

       2. unpow2 65.7%

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

       3. unpow2 65.7%

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

       4. swap-sqr 65.7%

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

       5. unpow2 65.7%

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

   16. Simplified 53.5%

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

Alternative 9: 65.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = powf(hypotf((floorf(w) * dY_46_u), t_0), 2.0f);
	float t_2 = dX_46_v * floorf(h);
	float t_3 = sqrtf(fmaxf(powf(hypotf(t_2, (dX_46_u * floorf(w))), 2.0f), t_1));
	float tmp;
	if (powf(t_2, 2.0f) >= t_1) {
		tmp = 1.0f / (t_3 / t_2);
	} else {
		tmp = t_0 / t_3;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0)
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = sqrt((((hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) != (hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0))) ? t_1 : ((t_1 != t_1) ? (hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) : max((hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)), t_1))))
	tmp = Float32(0.0)
	if ((t_2 ^ Float32(2.0)) >= t_1)
		tmp = Float32(Float32(1.0) / Float32(t_3 / t_2));
	else
		tmp = Float32(t_0 / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dY_46_v;
	t_1 = hypot((floor(w) * dY_46_u), t_0) ^ single(2.0);
	t_2 = dX_46_v * floor(h);
	t_3 = sqrt(max((hypot(t_2, (dX_46_u * floor(w))) ^ single(2.0)), t_1));
	tmp = single(0.0);
	if ((t_2 ^ single(2.0)) >= t_1)
		tmp = single(1.0) / (t_3 / t_2);
	else
		tmp = t_0 / t_3;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.5%

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

3. Simplified 78.7%

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

5. Taylor expanded in w around 0 78.5%

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

7. Simplified 78.3%

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

8. Taylor expanded in dX.u around 0 69.2%

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

   1. *-commutative 69.2%

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

   2. unpow2 69.2%

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

   3. unpow2 69.2%

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

   4. swap-sqr 69.2%

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

   5. unpow2 69.2%

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

   6. *-commutative 69.2%

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

10. Simplified 69.2%

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

11. Taylor expanded in dX.v around 0 69.3%

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

13. Simplified 69.7%

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

Alternative 10: 65.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.5%

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

3. Simplified 78.7%

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

5. Taylor expanded in w around 0 78.5%

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

7. Simplified 78.3%

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

8. Taylor expanded in dX.u around 0 69.2%

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

   1. *-commutative 69.2%

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

   2. unpow2 69.2%

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

   3. unpow2 69.2%

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

   4. swap-sqr 69.2%

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

   5. unpow2 69.2%

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

   6. *-commutative 69.2%

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

10. Simplified 69.2%

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

11. Taylor expanded in dX.v around 0 69.3%

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

13. Simplified 69.5%

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

Alternative 11: 65.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 78.5%

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

3. Simplified 78.7%

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

5. Taylor expanded in w around 0 78.5%

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

7. Simplified 78.3%

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

8. Taylor expanded in dX.u around 0 69.2%

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

   1. *-commutative 69.2%

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

   2. unpow2 69.2%

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

   3. unpow2 69.2%

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

   4. swap-sqr 69.2%

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

   5. unpow2 69.2%

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

   6. *-commutative 69.2%

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

10. Simplified 69.2%

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

11. Taylor expanded in dX.v around 0 69.3%

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

13. Simplified 69.5%

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

14. Taylor expanded in dX.v around inf 69.2%

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

    1. unpow2 69.2%

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

    2. unpow2 69.2%

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

    3. swap-sqr 69.2%

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

    4. unpow2 69.2%

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

16. Simplified 69.2%

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

Alternative 12: 62.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_3 := \sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_2, dX.u \cdot \left\lfloor w\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\right)}\\ t_4 := \frac{t\_1}{t\_3}\\ t_5 := dX.v \cdot \frac{\left\lfloor h\right\rfloor }{t\_3}\\ t_6 := {t\_2}^{2}\\ \mathbf{if}\;dY.v \leq -5000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_6 \geq {t\_1}^{2}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array}\\ \mathbf{elif}\;t\_6 \geq {t\_0}^{2}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* dX.v (floor h)))
        (t_3
         (sqrt
          (fmax
           (pow (hypot t_2 (* dX.u (floor w))) 2.0)
           (pow (hypot t_0 t_1) 2.0))))
        (t_4 (/ t_1 t_3))
        (t_5 (* dX.v (/ (floor h) t_3)))
        (t_6 (pow t_2 2.0)))
   (if (<= dY.v -5000.0)
     (if (>= t_6 (pow t_1 2.0)) t_5 t_4)
     (if (>= t_6 (pow t_0 2.0)) t_5 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(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = dX_46_v * floorf(h);
	float t_3 = sqrtf(fmaxf(powf(hypotf(t_2, (dX_46_u * floorf(w))), 2.0f), powf(hypotf(t_0, t_1), 2.0f)));
	float t_4 = t_1 / t_3;
	float t_5 = dX_46_v * (floorf(h) / t_3);
	float t_6 = powf(t_2, 2.0f);
	float tmp_1;
	if (dY_46_v <= -5000.0f) {
		float tmp_2;
		if (t_6 >= powf(t_1, 2.0f)) {
			tmp_2 = t_5;
		} else {
			tmp_2 = t_4;
		}
		tmp_1 = tmp_2;
	} else if (t_6 >= powf(t_0, 2.0f)) {
		tmp_1 = t_5;
	} else {
		tmp_1 = t_4;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = sqrt((((hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) != (hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? (hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) : max((hypot(t_2, Float32(dX_46_u * floor(w))) ^ Float32(2.0)), (hypot(t_0, t_1) ^ Float32(2.0))))))
	t_4 = Float32(t_1 / t_3)
	t_5 = Float32(dX_46_v * Float32(floor(h) / t_3))
	t_6 = t_2 ^ Float32(2.0)
	tmp_1 = Float32(0.0)
	if (dY_46_v <= Float32(-5000.0))
		tmp_2 = Float32(0.0)
		if (t_6 >= (t_1 ^ Float32(2.0)))
			tmp_2 = t_5;
		else
			tmp_2 = t_4;
		end
		tmp_1 = tmp_2;
	elseif (t_6 >= (t_0 ^ Float32(2.0)))
		tmp_1 = t_5;
	else
		tmp_1 = t_4;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = dX_46_v * floor(h);
	t_3 = sqrt(max((hypot(t_2, (dX_46_u * floor(w))) ^ single(2.0)), (hypot(t_0, t_1) ^ single(2.0))));
	t_4 = t_1 / t_3;
	t_5 = dX_46_v * (floor(h) / t_3);
	t_6 = t_2 ^ single(2.0);
	tmp_2 = single(0.0);
	if (dY_46_v <= single(-5000.0))
		tmp_3 = single(0.0);
		if (t_6 >= (t_1 ^ single(2.0)))
			tmp_3 = t_5;
		else
			tmp_3 = t_4;
		end
		tmp_2 = tmp_3;
	elseif (t_6 >= (t_0 ^ single(2.0)))
		tmp_2 = t_5;
	else
		tmp_2 = t_4;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < -5e3

   1. Initial program 65.5%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in w around 0 65.5%

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

   7. Simplified 65.1%

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

   8. Taylor expanded in dX.u around 0 61.7%

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

      1. *-commutative 61.7%

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

      2. unpow2 61.7%

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

      3. unpow2 61.7%

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

      4. swap-sqr 61.7%

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

      5. unpow2 61.7%

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

      6. *-commutative 61.7%

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

   10. Simplified 61.7%

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

   11. Taylor expanded in dX.v around 0 62.1%

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

   13. Simplified 62.4%

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

   14. Taylor expanded in dY.u around 0 62.4%

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

       1. *-commutative 62.4%

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

       2. unpow2 62.4%

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

       3. unpow2 62.4%

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

       4. swap-sqr 62.4%

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

       5. unpow2 62.4%

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

   16. Simplified 62.4%

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

   if -5e3 < dY.v

   1. Initial program 81.8%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in w around 0 81.8%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in dX.u around 0 71.1%

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

      1. *-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. unpow2 71.1%

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

      4. swap-sqr 71.1%

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

      5. unpow2 71.1%

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

      6. *-commutative 71.1%

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

   10. Simplified 71.1%

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

   11. Taylor expanded in dX.v around 0 71.2%

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

   13. Simplified 71.3%

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

   14. Taylor expanded in dY.u around inf 69.5%

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

       1. *-commutative 74.5%

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

       2. unpow2 74.5%

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

       3. unpow2 74.5%

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

       4. swap-sqr 74.5%

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

       5. unpow2 74.5%

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

   16. Simplified 69.5%

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

Alternative 13: 59.5% accurate, 1.3× speedup

Math

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

FPCore

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

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 = dX_46_v * floorf(h);
	float t_2 = sqrtf(fmaxf(powf(hypotf(t_1, (dX_46_u * floorf(w))), 2.0f), powf(hypotf((floorf(w) * dY_46_u), t_0), 2.0f)));
	float tmp;
	if (powf(t_1, 2.0f) >= powf(t_0, 2.0f)) {
		tmp = dX_46_v * (floorf(h) / t_2);
	} else {
		tmp = t_0 / t_2;
	}
	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(dX_46_v * floor(h))
	t_2 = sqrt((((hypot(t_1, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) != (hypot(t_1, Float32(dX_46_u * floor(w))) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0))) ? (hypot(t_1, Float32(dX_46_u * floor(w))) ^ Float32(2.0)) : max((hypot(t_1, Float32(dX_46_u * floor(w))) ^ Float32(2.0)), (hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0))))))
	tmp = Float32(0.0)
	if ((t_1 ^ Float32(2.0)) >= (t_0 ^ Float32(2.0)))
		tmp = Float32(dX_46_v * Float32(floor(h) / t_2));
	else
		tmp = Float32(t_0 / t_2);
	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 = dX_46_v * floor(h);
	t_2 = sqrt(max((hypot(t_1, (dX_46_u * floor(w))) ^ single(2.0)), (hypot((floor(w) * dY_46_u), t_0) ^ single(2.0))));
	tmp = single(0.0);
	if ((t_1 ^ single(2.0)) >= (t_0 ^ single(2.0)))
		tmp = dX_46_v * (floor(h) / t_2);
	else
		tmp = t_0 / t_2;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.5%

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

3. Simplified 78.7%

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

5. Taylor expanded in w around 0 78.5%

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

7. Simplified 78.3%

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

8. Taylor expanded in dX.u around 0 69.2%

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

   1. *-commutative 69.2%

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

   2. unpow2 69.2%

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

   3. unpow2 69.2%

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

   4. swap-sqr 69.2%

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

   5. unpow2 69.2%

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

   6. *-commutative 69.2%

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

10. Simplified 69.2%

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

11. Taylor expanded in dX.v around 0 69.3%

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

13. Simplified 69.5%

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

14. Taylor expanded in dY.u around 0 61.7%

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

    1. *-commutative 61.7%

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

    2. unpow2 61.7%

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

    3. unpow2 61.7%

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

    4. swap-sqr 61.7%

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

    5. unpow2 61.7%

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

16. Simplified 61.7%

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

Reproduce

herbie shell --seed 2024181 
(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))))