Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 75.9% → 76.0%

Time: 29.6s

Alternatives: 4

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: 75.9% accurate, 1.0× speedup

Math

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

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: 76.0% 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 73.8%

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

3. Simplified 74.0%

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

5. Final simplification 74.0%

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

Alternative 2: 76.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 73.8%

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

3. Simplified 74.0%

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

6. Applied egg-rr 73.9%

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

   1. associate-*l* 73.9%

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

   2. *-commutative 73.9%

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

8. Simplified 73.9%

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

9. Taylor expanded in w around 0 73.9%

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

11. Simplified 74.0%

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

12. Taylor expanded in w around 0 74.0%

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

    1. fma-udef 74.0%

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

    2. *-commutative 74.0%

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

    3. associate-*r* 74.0%

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

    4. unpow2 74.0%

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

    5. fma-def 74.0%

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

    6. unpow2 74.0%

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

    7. unpow2 74.0%

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

    8. swap-sqr 74.0%

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

    9. unpow2 74.0%

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

    10. fma-udef 74.0%

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

14. Simplified 74.0%

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

16. Applied egg-rr 74.0%

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

17. Final simplification 74.0%

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

Alternative 3: 75.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 73.8%

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

3. Simplified 74.0%

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

6. Applied egg-rr 73.9%

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

   1. associate-*l* 73.9%

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

   2. *-commutative 73.9%

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

8. Simplified 73.9%

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

9. Taylor expanded in w around 0 73.9%

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

11. Simplified 74.0%

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

12. Taylor expanded in w around 0 74.0%

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

    1. fma-udef 74.0%

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

    2. *-commutative 74.0%

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

    3. associate-*r* 74.0%

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

    4. unpow2 74.0%

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

    5. fma-def 74.0%

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

    6. unpow2 74.0%

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

    7. unpow2 74.0%

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

    8. swap-sqr 74.0%

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

    9. unpow2 74.0%

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

    10. fma-udef 74.0%

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

14. Simplified 74.0%

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

16. Applied egg-rr 74.0%

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

17. Final simplification 74.0%

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

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_1 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_2 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_3 := t_2 \cdot t_2\\ t_4 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_5 := t_4 \cdot t_4\\ t_6 := \frac{1}{\sqrt{\mathsf{max}\left(t_3 + t_0 \cdot t_0, t_5 + t_1 \cdot t_1\right)}}\\ \mathbf{if}\;t_3 + {t_0}^{2} \geq {t_1}^{2} + t_5:\\ \;\;\;\;t_0 \cdot t_6\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot t_6\\ \end{array} \end{array} \]

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 w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (* t_2 t_2))
        (t_4 (* (floor h) dY.v))
        (t_5 (* t_4 t_4))
        (t_6 (/ 1.0 (sqrt (fmax (+ t_3 (* t_0 t_0)) (+ t_5 (* t_1 t_1)))))))
   (if (>= (+ t_3 (pow t_0 2.0)) (+ (pow t_1 2.0) t_5))
     (* t_0 t_6)
     (* t_4 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(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = t_4 * t_4;
	float t_6 = 1.0f / sqrtf(fmaxf((t_3 + (t_0 * t_0)), (t_5 + (t_1 * t_1))));
	float tmp;
	if ((t_3 + powf(t_0, 2.0f)) >= (powf(t_1, 2.0f) + t_5)) {
		tmp = t_0 * t_6;
	} else {
		tmp = t_4 * t_6;
	}
	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(t_2 * t_2)
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(t_4 * t_4)
	t_6 = Float32(Float32(1.0) / sqrt(((Float32(t_3 + Float32(t_0 * t_0)) != Float32(t_3 + Float32(t_0 * t_0))) ? Float32(t_5 + Float32(t_1 * t_1)) : ((Float32(t_5 + Float32(t_1 * t_1)) != Float32(t_5 + Float32(t_1 * t_1))) ? Float32(t_3 + Float32(t_0 * t_0)) : max(Float32(t_3 + Float32(t_0 * t_0)), Float32(t_5 + Float32(t_1 * t_1)))))))
	tmp = Float32(0.0)
	if (Float32(t_3 + (t_0 ^ Float32(2.0))) >= Float32((t_1 ^ Float32(2.0)) + t_5))
		tmp = Float32(t_0 * t_6);
	else
		tmp = Float32(t_4 * t_6);
	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_4 = floor(h) * dY_46_v;
	t_5 = t_4 * t_4;
	t_6 = single(1.0) / sqrt(max((t_3 + (t_0 * t_0)), (t_5 + (t_1 * t_1))));
	tmp = single(0.0);
	if ((t_3 + (t_0 ^ single(2.0))) >= ((t_1 ^ single(2.0)) + t_5))
		tmp = t_0 * t_6;
	else
		tmp = t_4 * t_6;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 73.8%

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

   1. pow2 73.8%

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

4. Applied egg-rr 73.8%

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

5. Taylor expanded in w around 0 73.8%

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

   1. *-commutative 73.8%

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

   2. unpow2 73.8%

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

   3. unpow2 73.8%

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

   4. swap-sqr 73.8%

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

   5. unpow2 73.8%

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

7. Simplified 73.8%

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

8. Final simplification 73.8%

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

Reproduce

herbie shell --seed 2024024 
(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))))