Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 75.7% → 76.0%

Time: 24.8s

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: 75.7% 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: 76.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 78.5%

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

3. Simplified 78.6%

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

4. Taylor expanded in h around 0 78.6%

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

   1. *-commutative 78.6%

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

   2. unpow2 78.6%

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

   3. unpow2 78.6%

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

   4. swap-sqr 78.6%

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

   5. unpow2 78.6%

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

6. Simplified 78.6%

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

8. Applied egg-rr 78.8%

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

10. Applied egg-rr 74.5%

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

    1. expm1-def 76.2%

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot \left(dX.v \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right) \geq \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right):\\ \;\;\;\;\frac{1 \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot 1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\\ \end{array} \]

    2. expm1-log1p 78.8%

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

    3. *-commutative 78.8%

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

    4. *-commutative 78.8%

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

12. Simplified 78.8%

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

13. Final simplification 78.8%

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

Alternative 2: 75.7% accurate, 1.0× speedup

Math

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

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 w) dX.u)))
   (if (>=
        (fma
         (floor w)
         (* (floor w) (* dX.u dX.u))
         (* (floor h) (* (floor h) (* dX.v dX.v))))
        (fma
         (floor w)
         (* (floor w) (* dY.u dY.u))
         (* (floor h) (* (floor h) (* dY.v dY.v)))))
     (/
      t_1
      (pow
       (fmax
        (+ (* (floor h) (* (floor h) (pow dX.v 2.0))) (pow t_1 2.0))
        (pow t_0 2.0))
       0.5))
     (*
      (floor w)
      (*
       dY.u
       (pow
        (fmax
         (pow (* (floor h) dX.v) 2.0)
         (pow (hypot t_0 (* (floor h) dY.v)) 2.0))
        -0.5))))))

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(w) * dX_46_u;
	float tmp;
	if (fmaf(floorf(w), (floorf(w) * (dX_46_u * dX_46_u)), (floorf(h) * (floorf(h) * (dX_46_v * dX_46_v)))) >= fmaf(floorf(w), (floorf(w) * (dY_46_u * dY_46_u)), (floorf(h) * (floorf(h) * (dY_46_v * dY_46_v))))) {
		tmp = t_1 / powf(fmaxf(((floorf(h) * (floorf(h) * powf(dX_46_v, 2.0f))) + powf(t_1, 2.0f)), powf(t_0, 2.0f)), 0.5f);
	} else {
		tmp = floorf(w) * (dY_46_u * powf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_0, (floorf(h) * dY_46_v)), 2.0f)), -0.5f));
	}
	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(w) * dX_46_u)
	tmp = Float32(0.0)
	if (fma(floor(w), Float32(floor(w) * Float32(dX_46_u * dX_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dX_46_v * dX_46_v)))) >= fma(floor(w), Float32(floor(w) * Float32(dY_46_u * dY_46_u)), Float32(floor(h) * Float32(floor(h) * Float32(dY_46_v * dY_46_v)))))
		tmp = Float32(t_1 / (((Float32(Float32(floor(h) * Float32(floor(h) * (dX_46_v ^ Float32(2.0)))) + (t_1 ^ Float32(2.0))) != Float32(Float32(floor(h) * Float32(floor(h) * (dX_46_v ^ Float32(2.0)))) + (t_1 ^ Float32(2.0)))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? Float32(Float32(floor(h) * Float32(floor(h) * (dX_46_v ^ Float32(2.0)))) + (t_1 ^ Float32(2.0))) : max(Float32(Float32(floor(h) * Float32(floor(h) * (dX_46_v ^ Float32(2.0)))) + (t_1 ^ Float32(2.0))), (t_0 ^ Float32(2.0))))) ^ Float32(0.5)));
	else
		tmp = Float32(floor(w) * Float32(dY_46_u * ((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))))) ^ Float32(-0.5))));
	end
	return tmp
end

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Applied egg-rr 78.4%

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

7. Applied egg-rr 78.5%

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

8. Taylor expanded in dY.u around inf 78.5%

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

   1. *-commutative 78.5%

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

   2. unpow2 78.5%

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

   3. unpow2 78.5%

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

   4. swap-sqr 78.5%

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

   5. unpow2 78.5%

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

10. Simplified 78.5%

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

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

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

    1. unpow2 78.5%

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

    2. unpow2 78.5%

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

    3. swap-sqr 78.5%

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

    4. unpow2 78.5%

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

13. Simplified 78.5%

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

14. Final simplification 78.5%

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

Alternative 3: 75.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (* t_2 t_2))
        (t_4 (* (floor w) dX.u))
        (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 = floorf(h) * dX_46_v;
	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 = floorf(w) * dX_46_u;
	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(floor(h) * dX_46_v)
	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(floor(w) * dX_46_u)
	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 = floor(h) * dX_46_v;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(w) * dY_46_u;
	t_3 = t_2 * t_2;
	t_4 = floor(w) * dX_46_u;
	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 78.5%

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

   1. pow2 78.5%

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

3. Applied egg-rr 78.5%

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

   1. pow2 78.5%

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

5. Applied egg-rr 78.5%

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

6. Taylor expanded in h around 0 78.5%

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

   1. *-commutative 78.5%

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

   2. unpow2 78.5%

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

   3. unpow2 78.5%

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

   4. swap-sqr 78.5%

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

   5. unpow2 78.5%

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

8. Simplified 78.5%

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

9. Final simplification 78.5%

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

Reproduce

herbie shell --seed 2023298 
(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))))