Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 75.9% → 76.1%

Time: 31.6s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((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_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((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_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = (t_2 * t_2) + (t_0 * t_0);
	t_4 = floor(h) * dY_46_v;
	t_5 = (t_1 * t_1) + (t_4 * t_4);
	t_6 = single(1.0) / sqrt(max(t_3, t_5));
	tmp = single(0.0);
	if (t_3 >= t_5)
		tmp = t_6 * t_2;
	else
		tmp = t_6 * t_1;
	end
	tmp_2 = tmp;
end

Alternative 1: 76.1% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

6. Applied egg-rr 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \end{array} \]
7. Step-by-step derivation

   1. pow1/2 78.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\color{blue}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right)\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \end{array} \]

8. Applied egg-rr 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\color{blue}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \end{array} \]
9. Step-by-step derivation

   1. rem-cube-cbrt 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]

10. Applied egg-rr 79.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
11. Step-by-step derivation

    1. unpow1/2 79.4%

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

12. Applied egg-rr 79.5%

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

13. Final simplification 79.5%

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

Alternative 2: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := {\left(\mathsf{hypot}\left(t\_1, t\_0\right)\right)}^{2}\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_5 := {\left(\mathsf{hypot}\left(t\_3, t\_4\right)\right)}^{2}\\ \mathbf{if}\;t\_5 \geq {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}:\\ \;\;\;\;\frac{t\_3}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {t\_4}^{2}\right), t\_2\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\sqrt{\mathsf{max}\left(t\_5, t\_2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (pow (hypot t_1 t_0) 2.0))
        (t_3 (* (floor w) dX.u))
        (t_4 (* (floor h) dX.v))
        (t_5 (pow (hypot t_3 t_4) 2.0)))
   (if (>= t_5 (pow (hypot t_0 t_1) 2.0))
     (/
      t_3
      (pow
       (fmax (fma (floor w) (* (floor w) (pow dX.u 2.0)) (pow t_4 2.0)) t_2)
       0.5))
     (/ t_1 (sqrt (fmax t_5 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 = floorf(w) * dY_46_u;
	float t_2 = powf(hypotf(t_1, t_0), 2.0f);
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = powf(hypotf(t_3, t_4), 2.0f);
	float tmp;
	if (t_5 >= powf(hypotf(t_0, t_1), 2.0f)) {
		tmp = t_3 / powf(fmaxf(fmaf(floorf(w), (floorf(w) * powf(dX_46_u, 2.0f)), powf(t_4, 2.0f)), t_2), 0.5f);
	} else {
		tmp = t_1 / sqrtf(fmaxf(t_5, 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(floor(w) * dY_46_u)
	t_2 = hypot(t_1, t_0) ^ Float32(2.0)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = hypot(t_3, t_4) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_5 >= (hypot(t_0, t_1) ^ Float32(2.0)))
		tmp = Float32(t_3 / (((fma(floor(w), Float32(floor(w) * (dX_46_u ^ Float32(2.0))), (t_4 ^ Float32(2.0))) != fma(floor(w), Float32(floor(w) * (dX_46_u ^ Float32(2.0))), (t_4 ^ Float32(2.0)))) ? t_2 : ((t_2 != t_2) ? fma(floor(w), Float32(floor(w) * (dX_46_u ^ Float32(2.0))), (t_4 ^ Float32(2.0))) : max(fma(floor(w), Float32(floor(w) * (dX_46_u ^ Float32(2.0))), (t_4 ^ Float32(2.0))), t_2))) ^ Float32(0.5)));
	else
		tmp = Float32(t_1 / sqrt(((t_5 != t_5) ? t_2 : ((t_2 != t_2) ? t_5 : max(t_5, t_2)))));
	end
	return tmp
end

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

6. Applied egg-rr 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \end{array} \]
7. Step-by-step derivation

   1. pow1/2 78.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\color{blue}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right), \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right)\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \end{array} \]

8. Applied egg-rr 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\color{blue}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \end{array} \]
9. Step-by-step derivation

   1. rem-cube-cbrt 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]

10. Applied egg-rr 79.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor, \left\lfloorh\right\rfloor \cdot \left(\left\lfloorh\right\rfloor \cdot \left(dX.v \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorh\right\rfloor, dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right), dY.u \cdot \left(dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]

11. Taylor expanded in w around 0 79.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2} \geq {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]

13. Simplified 79.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]

14. Final simplification 79.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorh\right\rfloor \cdot dY.v, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dX.u}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor, \left\lfloorw\right\rfloor \cdot {dX.u}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
15. Add Preprocessing

Alternative 3: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := {\left(\mathsf{hypot}\left(t\_3, t\_0\right)\right)}^{2}\\ \mathbf{if}\;t\_3 \cdot t\_3 + t\_0 \cdot t\_0 \geq t\_1 \cdot t\_1 + t\_2 \cdot t\_2:\\ \;\;\;\;t\_3 \cdot \frac{1}{e^{\left(1.5 \cdot \log \left(\mathsf{max}\left(t\_4, {t\_1}^{2}\right)\right)\right) \cdot 0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t\_4, {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (hypot t_3 t_0) 2.0)))
   (if (>= (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2)))
     (*
      t_3
      (/
       1.0
       (exp (* (* 1.5 (log (fmax t_4 (pow t_1 2.0)))) 0.3333333333333333))))
     (* t_1 (/ 1.0 (sqrt (fmax t_4 (pow (hypot t_1 t_2) 2.0))))))))

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(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(hypotf(t_3, t_0), 2.0f);
	float tmp;
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2))) {
		tmp = t_3 * (1.0f / expf(((1.5f * logf(fmaxf(t_4, powf(t_1, 2.0f)))) * 0.3333333333333333f)));
	} else {
		tmp = t_1 * (1.0f / sqrtf(fmaxf(t_4, powf(hypotf(t_1, t_2), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = hypot(t_3, t_0) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) >= Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)))
		tmp = Float32(t_3 * Float32(Float32(1.0) / exp(Float32(Float32(Float32(1.5) * log(((t_4 != t_4) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? t_4 : max(t_4, (t_1 ^ Float32(2.0))))))) * Float32(0.3333333333333333)))));
	else
		tmp = Float32(t_1 * Float32(Float32(1.0) / sqrt(((t_4 != t_4) ? (hypot(t_1, t_2) ^ Float32(2.0)) : (((hypot(t_1, t_2) ^ Float32(2.0)) != (hypot(t_1, t_2) ^ Float32(2.0))) ? t_4 : max(t_4, (hypot(t_1, t_2) ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(w) * dX_46_u;
	t_4 = hypot(t_3, t_0) ^ single(2.0);
	tmp = single(0.0);
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2)))
		tmp = t_3 * (single(1.0) / exp(((single(1.5) * log(max(t_4, (t_1 ^ single(2.0))))) * single(0.3333333333333333))));
	else
		tmp = t_1 * (single(1.0) / sqrt(max(t_4, (hypot(t_1, t_2) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 79.2%

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

4. Applied egg-rr 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{1.5}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Taylor expanded in dY.u around inf 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
6. Step-by-step derivation

   1. *-commutative 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dY.u}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. swap-sqr 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Simplified 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Taylor expanded in w around 0 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
9. Step-by-step derivation

10. Simplified 60.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
11. Step-by-step derivation

    1. pow1/3 58.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{{\left({\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    2. pow-to-exp 58.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{e^{\log \left({\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}\right) \cdot 0.3333333333333333}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    3. pow2 58.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{e^{\log \left({\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)}\right)\right)}^{1.5}\right) \cdot 0.3333333333333333}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    4. log-pow 75.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{e^{\color{blue}{\left(1.5 \cdot \log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\right)\right)\right)} \cdot 0.3333333333333333}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    5. pow2 75.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{e^{\left(1.5 \cdot \log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)\right) \cdot 0.3333333333333333}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

12. Applied egg-rr 75.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{e^{\left(1.5 \cdot \log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)\right) \cdot 0.3333333333333333}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

13. Final simplification 75.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{e^{\left(1.5 \cdot \log \left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)\right) \cdot 0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
14. Add Preprocessing

Alternative 4: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := {\left(\mathsf{hypot}\left(t\_3, t\_0\right)\right)}^{2}\\ \mathbf{if}\;t\_3 \cdot t\_3 + t\_0 \cdot t\_0 \geq t\_1 \cdot t\_1 + t\_2 \cdot t\_2:\\ \;\;\;\;t\_3 \cdot \frac{1}{{\left(\sqrt[3]{\sqrt{\mathsf{max}\left(t\_4, {t\_1}^{2}\right)}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t\_4, {\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (hypot t_3 t_0) 2.0)))
   (if (>= (+ (* t_3 t_3) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2)))
     (* t_3 (/ 1.0 (pow (cbrt (sqrt (fmax t_4 (pow t_1 2.0)))) 3.0)))
     (* t_1 (/ 1.0 (sqrt (fmax t_4 (pow (hypot t_1 t_2) 2.0))))))))

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(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(hypotf(t_3, t_0), 2.0f);
	float tmp;
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2))) {
		tmp = t_3 * (1.0f / powf(cbrtf(sqrtf(fmaxf(t_4, powf(t_1, 2.0f)))), 3.0f));
	} else {
		tmp = t_1 * (1.0f / sqrtf(fmaxf(t_4, powf(hypotf(t_1, t_2), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = hypot(t_3, t_0) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) >= Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)))
		tmp = Float32(t_3 * Float32(Float32(1.0) / (cbrt(sqrt(((t_4 != t_4) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? t_4 : max(t_4, (t_1 ^ Float32(2.0))))))) ^ Float32(3.0))));
	else
		tmp = Float32(t_1 * Float32(Float32(1.0) / sqrt(((t_4 != t_4) ? (hypot(t_1, t_2) ^ Float32(2.0)) : (((hypot(t_1, t_2) ^ Float32(2.0)) != (hypot(t_1, t_2) ^ Float32(2.0))) ? t_4 : max(t_4, (hypot(t_1, t_2) ^ Float32(2.0))))))));
	end
	return tmp
end

Derivation

1. Initial program 79.2%

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

4. Applied egg-rr 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{1.5}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Taylor expanded in dY.u around inf 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
6. Step-by-step derivation

   1. *-commutative 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dY.u}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. swap-sqr 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Simplified 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Taylor expanded in w around 0 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
9. Step-by-step derivation

10. Simplified 60.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
11. Step-by-step derivation

    1. add-cube-cbrt 59.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    2. pow3 60.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}}\right)}^{3}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

12. Applied egg-rr 78.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}\right)}^{3}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

13. Final simplification 78.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{{\left(\sqrt[3]{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
14. Add Preprocessing

Alternative 5: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.2%

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

   1. pow2 79.2%

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

4. Applied egg-rr 79.2%

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

   1. pow2 79.2%

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

6. Applied egg-rr 79.2%

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

7. Taylor expanded in w around 0 79.2%

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

9. Simplified 79.2%

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

10. Final simplification 79.2%

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

Alternative 6: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {t\_2}^{2}\\ t_4 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_5 := {\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}\\ \mathbf{if}\;t\_4 \cdot t\_4 + t\_0 \cdot t\_0 \geq t\_1 \cdot t\_1 + t\_3:\\ \;\;\;\;t\_4 \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{max}\left(t\_5, t\_3\right)\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{1}{\sqrt{\mathsf{max}\left(t\_5, {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow t_2 2.0))
        (t_4 (* (floor w) dX.u))
        (t_5 (pow (hypot t_4 t_0) 2.0)))
   (if (>= (+ (* t_4 t_4) (* t_0 t_0)) (+ (* t_1 t_1) t_3))
     (* t_4 (/ 1.0 (cbrt (pow (fmax t_5 t_3) 1.5))))
     (* t_2 (/ 1.0 (sqrt (fmax t_5 (pow (hypot t_2 t_1) 2.0))))))))

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(h) * dX_46_v;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(t_2, 2.0f);
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = powf(hypotf(t_4, t_0), 2.0f);
	float tmp;
	if (((t_4 * t_4) + (t_0 * t_0)) >= ((t_1 * t_1) + t_3)) {
		tmp = t_4 * (1.0f / cbrtf(powf(fmaxf(t_5, t_3), 1.5f)));
	} else {
		tmp = t_2 * (1.0f / sqrtf(fmaxf(t_5, powf(hypotf(t_2, t_1), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = t_2 ^ Float32(2.0)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = hypot(t_4, t_0) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_4 * t_4) + Float32(t_0 * t_0)) >= Float32(Float32(t_1 * t_1) + t_3))
		tmp = Float32(t_4 * Float32(Float32(1.0) / cbrt((((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3))) ^ Float32(1.5)))));
	else
		tmp = Float32(t_2 * Float32(Float32(1.0) / sqrt(((t_5 != t_5) ? (hypot(t_2, t_1) ^ Float32(2.0)) : (((hypot(t_2, t_1) ^ Float32(2.0)) != (hypot(t_2, t_1) ^ Float32(2.0))) ? t_5 : max(t_5, (hypot(t_2, t_1) ^ Float32(2.0))))))));
	end
	return tmp
end

Derivation

1. Initial program 79.2%

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

4. Applied egg-rr 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\color{blue}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{1.5}}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

5. Taylor expanded in dY.u around inf 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
6. Step-by-step derivation

   1. *-commutative 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dY.u}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. swap-sqr 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow2 60.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Simplified 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right), \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

8. Taylor expanded in w around 0 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
9. Step-by-step derivation

10. Simplified 60.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

11. Taylor expanded in w around 0 60.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]
12. Step-by-step derivation

    1. *-commutative 60.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \color{blue}{{\left(\left\lfloorw\right\rfloor\right)}^{2} \cdot {dY.u}^{2}} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    2. unpow2 60.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right)} \cdot {dY.u}^{2} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    3. unpow2 60.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    4. swap-sqr 60.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    5. unpow2 60.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

13. Simplified 60.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \color{blue}{{\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}} + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}} \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}} \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right)\\ \end{array} \]

14. Final simplification 60.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right) + {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dX.u)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dY.u))))