Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 76.4% → 76.7%

Time: 23.0s

Alternatives: 3

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: 76.4% 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: 76.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.1%

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

5. Applied egg-rr 75.0%

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

   1. expm1-log1p-u 72.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}\right)\right)}}\\ \end{array} \]

   2. expm1-udef 71.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}\right)} - 1}}\\ \end{array} \]

7. Applied egg-rr 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot {dX.v}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)} - 1}}\\ \end{array} \]
8. Step-by-step derivation

   1. expm1-def 72.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot {dX.v}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\right)}}\\ \end{array} \]

   2. expm1-log1p 75.1%

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

9. Simplified 75.1%

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

11. Applied egg-rr 75.3%

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

12. Taylor expanded in h around 0 75.3%

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

    1. fma-udef 75.3%

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

    2. *-commutative 75.3%

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

    3. associate-*r* 75.3%

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

    4. unpow2 75.3%

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

    5. *-commutative 75.3%

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

    6. unpow2 75.3%

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

    7. unpow2 75.3%

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

    8. swap-sqr 75.3%

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

    9. unpow2 75.3%

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

    10. *-commutative 75.3%

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

    11. unpow2 75.3%

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

    12. unpow2 75.3%

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

    13. swap-sqr 75.3%

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

    14. unpow2 75.3%

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

14. Simplified 75.3%

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

15. Final simplification 75.3%

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

Alternative 2: 76.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.v (floor h)))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (pow (hypot t_1 t_2) 2.0))
        (t_4 (* dX.u (floor w)))
        (t_5 (+ (pow t_4 2.0) (pow t_0 2.0))))
   (if (>= t_5 (fma (pow (floor w) 2.0) (pow dY.u 2.0) (pow t_2 2.0)))
     (* dX.u (* (floor w) (pow (fmax (pow (hypot t_0 t_4) 2.0) t_3) -0.5)))
     (/ t_1 (sqrt (fmax t_5 t_3))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dX_46_v * floorf(h);
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = powf(hypotf(t_1, t_2), 2.0f);
	float t_4 = dX_46_u * floorf(w);
	float t_5 = powf(t_4, 2.0f) + powf(t_0, 2.0f);
	float tmp;
	if (t_5 >= fmaf(powf(floorf(w), 2.0f), powf(dY_46_u, 2.0f), powf(t_2, 2.0f))) {
		tmp = dX_46_u * (floorf(w) * powf(fmaxf(powf(hypotf(t_0, t_4), 2.0f), t_3), -0.5f));
	} else {
		tmp = t_1 / sqrtf(fmaxf(t_5, t_3));
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dX_46_v * floor(h))
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = hypot(t_1, t_2) ^ Float32(2.0)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = Float32((t_4 ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (t_5 >= fma((floor(w) ^ Float32(2.0)), (dY_46_u ^ Float32(2.0)), (t_2 ^ Float32(2.0))))
		tmp = Float32(dX_46_u * Float32(floor(w) * ((((hypot(t_0, t_4) ^ Float32(2.0)) != (hypot(t_0, t_4) ^ Float32(2.0))) ? t_3 : ((t_3 != t_3) ? (hypot(t_0, t_4) ^ Float32(2.0)) : max((hypot(t_0, t_4) ^ Float32(2.0)), t_3))) ^ Float32(-0.5))));
	else
		tmp = Float32(t_1 / sqrt(((t_5 != t_5) ? t_3 : ((t_3 != t_3) ? t_5 : max(t_5, t_3)))));
	end
	return tmp
end

Derivation

1. Initial program 75.0%

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

3. Simplified 75.1%

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

5. Applied egg-rr 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ \end{array} \]
6. Step-by-step derivation

   1. expm1-def 75.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}}\\ \end{array} \]

   2. expm1-log1p 75.2%

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

   3. associate-/l* 75.2%

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

   4. *-commutative 75.2%

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

   5. *-commutative 75.2%

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

7. Simplified 75.2%

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

   1. expm1-log1p-u 72.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}\right)\right)}}\\ \end{array} \]

   2. expm1-udef 71.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right)\right)}\right)} - 1}}\\ \end{array} \]

9. Applied egg-rr 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;\frac{\left\lfloorw\right\rfloor}{\frac{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}{dX.u}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot {dX.v}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)} - 1}}\\ \end{array} \]
10. Step-by-step derivation

    1. expm1-def 72.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left(dX.v \cdot dX.v\right) \cdot \left\lfloorh\right\rfloor, \left\lfloorw\right\rfloor \cdot \left(\left(dX.u \cdot dX.u\right) \cdot \left\lfloorw\right\rfloor\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor, dY.u \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right), \left\lfloorh\right\rfloor \cdot \left(\left(dY.v \cdot dY.v\right) \cdot \left\lfloorh\right\rfloor\right)\right):\\ \;\;\;\;{\left(\sqrt[3]{\frac{\left\lfloorw\right\rfloor \cdot dX.u}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\left\lfloorw\right\rfloor \cdot dY.u}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorh\right\rfloor, \left\lfloorh\right\rfloor \cdot {dX.v}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\right)}}\\ \end{array} \]

    2. expm1-log1p 75.1%

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

11. Simplified 75.3%

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

13. Applied egg-rr 75.1%

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

    1. *-commutative 75.1%

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

    2. associate-*l* 75.2%

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

    3. *-commutative 75.2%

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

15. Simplified 75.2%

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

16. Taylor expanded in h around 0 75.2%

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

    1. fma-udef 75.3%

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

    2. *-commutative 75.3%

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

    3. associate-*r* 75.3%

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

    4. unpow2 75.3%

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

    5. *-commutative 75.3%

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

    6. unpow2 75.3%

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

    7. unpow2 75.3%

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

    8. swap-sqr 75.3%

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

    9. unpow2 75.3%

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

    10. *-commutative 75.3%

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

    11. unpow2 75.3%

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

    12. unpow2 75.3%

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

    13. swap-sqr 75.3%

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

    14. unpow2 75.3%

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

18. Simplified 75.2%

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

19. Final simplification 75.2%

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

Alternative 3: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 75.0%

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

   1. pow2 75.0%

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

3. Applied egg-rr 75.0%

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

   1. pow2 75.0%

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

5. Applied egg-rr 75.0%

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

6. Taylor expanded in h around 0 75.0%

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

   1. *-commutative 75.0%

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

   2. unpow2 75.0%

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

   3. unpow2 75.0%

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

   4. swap-sqr 75.0%

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

   5. unpow2 75.0%

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

8. Simplified 75.0%

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

9. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2023322 
(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))))