Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.7% → 76.8%

Time: 30.9s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5)))))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_0);
	else
		tmp = Float32(t_6 * t_4);
	end
	return tmp
end

MATLAB

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((t_3 != t_3) ? t_5 : ((t_5 != t_5) ? t_3 : max(t_3, t_5)))))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32(t_6 * t_0);
	else
		tmp = Float32(t_6 * t_4);
	end
	return tmp
end

MATLAB

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

Alternative 1: 76.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.9%

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

3. Simplified 80.1%

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

5. Final simplification 80.1%

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

Alternative 2: 76.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.9%

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

3. Simplified 80.0%

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

   1. pow2 80.0%

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

6. Applied egg-rr 80.0%

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

7. Taylor expanded in w around 0 80.0%

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

9. Simplified 80.0%

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

10. Final simplification 80.0%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.9%

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

3. Taylor expanded in w around 0 80.0%

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

   1. pow2 80.0%

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

5. Applied egg-rr 80.0%

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

6. Taylor expanded in w around 0 80.0%

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

8. Simplified 80.0%

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

9. Final simplification 80.0%

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

Alternative 4: 76.7% accurate, 1.0× speedup

Math

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

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 (* t_1 t_1))
        (t_3 (* (floor w) dY.u))
        (t_4 (* (floor w) dX.u))
        (t_5
         (/
          1.0
          (sqrt (fmax (+ (* t_0 t_0) (* t_4 t_4)) (+ (* t_3 t_3) t_2))))))
   (if (>= (+ (pow t_4 2.0) (pow t_0 2.0)) (+ t_2 (pow t_3 2.0)))
     (* t_0 t_5)
     (* t_1 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 = t_1 * t_1;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = 1.0f / sqrtf(fmaxf(((t_0 * t_0) + (t_4 * t_4)), ((t_3 * t_3) + t_2)));
	float tmp;
	if ((powf(t_4, 2.0f) + powf(t_0, 2.0f)) >= (t_2 + powf(t_3, 2.0f))) {
		tmp = t_0 * t_5;
	} else {
		tmp = t_1 * 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(t_1 * t_1)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(Float32(1.0) / sqrt(((Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)) != Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4))) ? Float32(Float32(t_3 * t_3) + t_2) : ((Float32(Float32(t_3 * t_3) + t_2) != Float32(Float32(t_3 * t_3) + t_2)) ? Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)) : max(Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)), Float32(Float32(t_3 * t_3) + t_2))))))
	tmp = Float32(0.0)
	if (Float32((t_4 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) >= Float32(t_2 + (t_3 ^ Float32(2.0))))
		tmp = Float32(t_0 * t_5);
	else
		tmp = Float32(t_1 * 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 = t_1 * t_1;
	t_3 = floor(w) * dY_46_u;
	t_4 = floor(w) * dX_46_u;
	t_5 = single(1.0) / sqrt(max(((t_0 * t_0) + (t_4 * t_4)), ((t_3 * t_3) + t_2)));
	tmp = single(0.0);
	if (((t_4 ^ single(2.0)) + (t_0 ^ single(2.0))) >= (t_2 + (t_3 ^ single(2.0))))
		tmp = t_0 * t_5;
	else
		tmp = t_1 * t_5;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 79.9%

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

   1. pow2 79.9%

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

4. Applied egg-rr 79.9%

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

   1. pow2 80.0%

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

6. Applied egg-rr 79.9%

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

7. Taylor expanded in w around 0 79.9%

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

   1. *-commutative 79.9%

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

   2. unpow2 79.9%

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

   3. unpow2 79.9%

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

   4. swap-sqr 79.9%

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

   5. unpow2 79.9%

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

9. Simplified 79.9%

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

10. Final simplification 79.9%

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

Alternative 5: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_3 := {\left(\mathsf{hypot}\left(t\_2, t\_1\right)\right)}^{2}\\ t_4 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_5 := \sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_4, t\_4, t\_0 \cdot t\_0\right), \mathsf{fma}\left(t\_2, t\_2, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t\_1\right)\right)\right)}\\ t_6 := t\_1 \cdot \frac{1}{t\_5}\\ t_7 := {\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}\\ \mathbf{if}\;dX.u \leq 0.05999999865889549:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_7 \geq {t\_2}^{2}:\\ \;\;\;\;\frac{t\_0}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array}\\ \mathbf{elif}\;{t\_4}^{2} \geq t\_3:\\ \;\;\;\;\frac{t\_0}{{\left(\mathsf{max}\left(t\_7, t\_3\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow (hypot t_2 t_1) 2.0))
        (t_4 (* (floor w) dX.u))
        (t_5
         (sqrt
          (fmax
           (fma t_4 t_4 (* t_0 t_0))
           (fma t_2 t_2 (* (floor h) (* dY.v t_1))))))
        (t_6 (* t_1 (/ 1.0 t_5)))
        (t_7 (pow (hypot t_4 t_0) 2.0)))
   (if (<= dX.u 0.05999999865889549)
     (if (>= t_7 (pow t_2 2.0)) (/ t_0 t_5) t_6)
     (if (>= (pow t_4 2.0) t_3) (/ t_0 (pow (fmax t_7 t_3) 0.5)) t_6))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(hypotf(t_2, t_1), 2.0f);
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = sqrtf(fmaxf(fmaf(t_4, t_4, (t_0 * t_0)), fmaf(t_2, t_2, (floorf(h) * (dY_46_v * t_1)))));
	float t_6 = t_1 * (1.0f / t_5);
	float t_7 = powf(hypotf(t_4, t_0), 2.0f);
	float tmp_1;
	if (dX_46_u <= 0.05999999865889549f) {
		float tmp_2;
		if (t_7 >= powf(t_2, 2.0f)) {
			tmp_2 = t_0 / t_5;
		} else {
			tmp_2 = t_6;
		}
		tmp_1 = tmp_2;
	} else if (powf(t_4, 2.0f) >= t_3) {
		tmp_1 = t_0 / powf(fmaxf(t_7, t_3), 0.5f);
	} else {
		tmp_1 = t_6;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = hypot(t_2, t_1) ^ Float32(2.0)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = sqrt(((fma(t_4, t_4, Float32(t_0 * t_0)) != fma(t_4, t_4, Float32(t_0 * t_0))) ? fma(t_2, t_2, Float32(floor(h) * Float32(dY_46_v * t_1))) : ((fma(t_2, t_2, Float32(floor(h) * Float32(dY_46_v * t_1))) != fma(t_2, t_2, Float32(floor(h) * Float32(dY_46_v * t_1)))) ? fma(t_4, t_4, Float32(t_0 * t_0)) : max(fma(t_4, t_4, Float32(t_0 * t_0)), fma(t_2, t_2, Float32(floor(h) * Float32(dY_46_v * t_1)))))))
	t_6 = Float32(t_1 * Float32(Float32(1.0) / t_5))
	t_7 = hypot(t_4, t_0) ^ Float32(2.0)
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(0.05999999865889549))
		tmp_2 = Float32(0.0)
		if (t_7 >= (t_2 ^ Float32(2.0)))
			tmp_2 = Float32(t_0 / t_5);
		else
			tmp_2 = t_6;
		end
		tmp_1 = tmp_2;
	elseif ((t_4 ^ Float32(2.0)) >= t_3)
		tmp_1 = Float32(t_0 / (((t_7 != t_7) ? t_3 : ((t_3 != t_3) ? t_7 : max(t_7, t_3))) ^ Float32(0.5)));
	else
		tmp_1 = t_6;
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.0599999987

   1. Initial program 81.2%

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

   3. Simplified 81.3%

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

      1. pow2 81.3%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in w around 0 81.3%

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

   9. Simplified 81.3%

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

   10. Taylor expanded in dY.u around inf 71.7%

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

       1. *-commutative 71.7%

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

       2. unpow2 71.7%

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

       3. unpow2 71.7%

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

       4. swap-sqr 71.7%

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

       5. unpow2 71.7%

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

   12. Simplified 71.7%

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

   if 0.0599999987 < dX.u

   1. Initial program 76.3%

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

   3. Simplified 76.3%

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

      1. pow2 76.3%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in w around 0 76.3%

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

   9. Simplified 76.3%

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

   10. Taylor expanded in dX.u around inf 76.3%

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

       1. unpow2 76.3%

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

       2. unpow2 76.3%

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

       3. swap-sqr 76.3%

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

       4. unpow2 76.3%

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

   12. Simplified 76.3%

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

       1. pow1/2 76.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{1 \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}{\color{blue}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]

   14. Applied egg-rr 76.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{1 \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}{\color{blue}{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 0.05999999865889549:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2} \geq {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dX.v}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\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\lfloorh\right\rfloor \cdot dY.v\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(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\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}\\ \mathbf{elif}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dX.v}{{\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}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorh\right\rfloor \cdot dY.v\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(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\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. Add Preprocessing

Alternative 6: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\\ \mathbf{if}\;{t\_3}^{2} \geq t\_4:\\ \;\;\;\;\frac{t\_2}{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_3, t\_2\right)\right)}^{2}, t\_4\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2\right), \mathsf{fma}\left(t\_0, t\_0, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t\_1\right)\right)\right)}}\\ \end{array} \end{array} \]

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 (* (floor h) dX.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (hypot t_0 t_1) 2.0)))
   (if (>= (pow t_3 2.0) t_4)
     (/ t_2 (pow (fmax (pow (hypot t_3 t_2) 2.0) t_4) 0.5))
     (*
      t_1
      (/
       1.0
       (sqrt
        (fmax
         (fma t_3 t_3 (* t_2 t_2))
         (fma t_0 t_0 (* (floor h) (* dY.v 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(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(hypotf(t_0, t_1), 2.0f);
	float tmp;
	if (powf(t_3, 2.0f) >= t_4) {
		tmp = t_2 / powf(fmaxf(powf(hypotf(t_3, t_2), 2.0f), t_4), 0.5f);
	} else {
		tmp = t_1 * (1.0f / sqrtf(fmaxf(fmaf(t_3, t_3, (t_2 * t_2)), fmaf(t_0, t_0, (floorf(h) * (dY_46_v * 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(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = hypot(t_0, t_1) ^ Float32(2.0)
	tmp = Float32(0.0)
	if ((t_3 ^ Float32(2.0)) >= t_4)
		tmp = Float32(t_2 / ((((hypot(t_3, t_2) ^ Float32(2.0)) != (hypot(t_3, t_2) ^ Float32(2.0))) ? t_4 : ((t_4 != t_4) ? (hypot(t_3, t_2) ^ Float32(2.0)) : max((hypot(t_3, t_2) ^ Float32(2.0)), t_4))) ^ Float32(0.5)));
	else
		tmp = Float32(t_1 * Float32(Float32(1.0) / sqrt(((fma(t_3, t_3, Float32(t_2 * t_2)) != fma(t_3, t_3, Float32(t_2 * t_2))) ? fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) : ((fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) != fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1)))) ? fma(t_3, t_3, Float32(t_2 * t_2)) : max(fma(t_3, t_3, Float32(t_2 * t_2)), fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1)))))))));
	end
	return tmp
end

Derivation

1. Initial program 79.9%

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

3. Simplified 80.0%

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

   1. pow2 80.0%

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

6. Applied egg-rr 80.0%

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

7. Taylor expanded in w around 0 80.0%

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

9. Simplified 80.0%

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

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

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

    1. unpow2 68.6%

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

    2. unpow2 68.6%

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

    3. swap-sqr 68.6%

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

    4. unpow2 68.6%

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

12. Simplified 68.6%

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

    1. pow1/2 68.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{1 \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}{\color{blue}{{\left(\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]

14. Applied egg-rr 68.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{1 \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)}{\color{blue}{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]

15. Final simplification 68.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2} \geq {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}:\\ \;\;\;\;\frac{\left\lfloorh\right\rfloor \cdot dX.v}{{\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorw\right\rfloor \cdot dX.u, \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)\right)}}\\ \end{array} \]
16. Add Preprocessing

Alternative 7: 65.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.9%

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

3. Simplified 80.0%

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

   1. pow2 80.0%

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

6. Applied egg-rr 80.0%

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

7. Taylor expanded in w around 0 80.0%

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

9. Simplified 80.0%

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

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

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

    1. unpow2 68.6%

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

    2. unpow2 68.6%

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

    3. swap-sqr 68.6%

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

    4. unpow2 68.6%

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

12. Simplified 68.6%

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

13. Final simplification 68.6%

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

Alternative 8: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\\ \mathbf{if}\;{t\_3}^{2} \geq t\_4:\\ \;\;\;\;\frac{t\_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_3, t\_2\right)\right)}^{2}, t\_4\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{1}{\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2\right), \mathsf{fma}\left(t\_0, t\_0, \left\lfloorh\right\rfloor \cdot \left(dY.v \cdot t\_1\right)\right)\right)}}\\ \end{array} \end{array} \]

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 (* (floor h) dX.v))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (hypot t_0 t_1) 2.0)))
   (if (>= (pow t_3 2.0) t_4)
     (/ t_2 (expm1 (log1p (sqrt (fmax (pow (hypot t_3 t_2) 2.0) t_4)))))
     (*
      t_1
      (/
       1.0
       (sqrt
        (fmax
         (fma t_3 t_3 (* t_2 t_2))
         (fma t_0 t_0 (* (floor h) (* dY.v 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(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(hypotf(t_0, t_1), 2.0f);
	float tmp;
	if (powf(t_3, 2.0f) >= t_4) {
		tmp = t_2 / expm1f(log1pf(sqrtf(fmaxf(powf(hypotf(t_3, t_2), 2.0f), t_4))));
	} else {
		tmp = t_1 * (1.0f / sqrtf(fmaxf(fmaf(t_3, t_3, (t_2 * t_2)), fmaf(t_0, t_0, (floorf(h) * (dY_46_v * 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(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = hypot(t_0, t_1) ^ Float32(2.0)
	tmp = Float32(0.0)
	if ((t_3 ^ Float32(2.0)) >= t_4)
		tmp = Float32(t_2 / expm1(log1p(sqrt((((hypot(t_3, t_2) ^ Float32(2.0)) != (hypot(t_3, t_2) ^ Float32(2.0))) ? t_4 : ((t_4 != t_4) ? (hypot(t_3, t_2) ^ Float32(2.0)) : max((hypot(t_3, t_2) ^ Float32(2.0)), t_4)))))));
	else
		tmp = Float32(t_1 * Float32(Float32(1.0) / sqrt(((fma(t_3, t_3, Float32(t_2 * t_2)) != fma(t_3, t_3, Float32(t_2 * t_2))) ? fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) : ((fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1))) != fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1)))) ? fma(t_3, t_3, Float32(t_2 * t_2)) : max(fma(t_3, t_3, Float32(t_2 * t_2)), fma(t_0, t_0, Float32(floor(h) * Float32(dY_46_v * t_1)))))))));
	end
	return tmp
end

Derivation

1. Initial program 79.9%

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

3. Simplified 80.0%

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

   1. pow2 80.0%

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

6. Applied egg-rr 80.0%

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

7. Taylor expanded in w around 0 80.0%

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

9. Simplified 80.0%

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

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

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

    1. unpow2 68.6%

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

    2. unpow2 68.6%

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

    3. swap-sqr 68.6%

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

    4. unpow2 68.6%

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

12. Simplified 68.6%

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

14. Applied egg-rr 66.9%

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

15. Final simplification 66.9%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, v)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dX.v)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dY.v))))