Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 75.4% → 75.5%

Time: 18.4s

Alternatives: 10

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 75.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* t_1 dY.v))
        (t_3 (pow (floor w) 2.0))
        (t_4 (fma t_2 dY.v (* (* dY.u dY.u) t_3)))
        (t_5 (fma (* dX.u dX.u) t_3 (* (* t_1 dX.v) dX.v))))
   (if (>= t_5 t_4)
     (/ (* (floor w) dX.u) (sqrt (fmax t_5 t_4)))
     (/ t_0 (sqrt (fmax t_5 (fma t_2 dY.v (pow t_0 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(w) * dY_46_u;
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = t_1 * dY_46_v;
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = fmaf(t_2, dY_46_v, ((dY_46_u * dY_46_u) * t_3));
	float t_5 = fmaf((dX_46_u * dX_46_u), t_3, ((t_1 * dX_46_v) * dX_46_v));
	float tmp;
	if (t_5 >= t_4) {
		tmp = (floorf(w) * dX_46_u) / sqrtf(fmaxf(t_5, t_4));
	} else {
		tmp = t_0 / sqrtf(fmaxf(t_5, fmaf(t_2, dY_46_v, powf(t_0, 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(w) * dY_46_u)
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(t_1 * dY_46_v)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = fma(t_2, dY_46_v, Float32(Float32(dY_46_u * dY_46_u) * t_3))
	t_5 = fma(Float32(dX_46_u * dX_46_u), t_3, Float32(Float32(t_1 * dX_46_v) * dX_46_v))
	tmp = Float32(0.0)
	if (t_5 >= t_4)
		tmp = Float32(Float32(floor(w) * dX_46_u) / sqrt(fmax(t_5, t_4)));
	else
		tmp = Float32(t_0 / sqrt(fmax(t_5, fma(t_2, dY_46_v, (t_0 ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Initial program 76.8%

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

   1. lift-floor.f32 N/A

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

   2. unpow1 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. lift-floor.f32 76.1

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

4. Applied rewrites 76.1%

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

6. Applied rewrites 76.8%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-floor.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. lift-floor.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-pow.f32 76.9

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

   11. *-lft-identity N/A

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

   12. lift-*.f32 N/A

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

   13. lift-floor.f32 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift-floor.f32 76.9

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

8. Applied rewrites 76.9%

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

9. Final simplification 76.9%

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

Alternative 2: 75.3% accurate, N/A× speedup

Math

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

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 (pow (floor h) 2.0))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (floor w) 2.0))
        (t_5 (+ (* t_3 t_3) (* t_0 t_0)))
        (t_6 (* (floor h) dY.v))
        (t_7 (+ (* t_1 t_1) (* t_6 t_6))))
   (if (>= t_5 t_7)
     (* (/ 1.0 (sqrt (fmax t_5 t_7))) t_3)
     (*
      (*
       (pow
        (fmax
         (- (* (* t_2 dX.v) dX.v) (* (* -1.0 (* dX.u t_4)) dX.u))
         (- (* (* t_2 dY.v) dY.v) (* (* -1.0 (* dY.u t_4)) dY.u)))
        -0.5)
       dY.u)
      (floor w)))))

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 = powf(floorf(h), 2.0f);
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = (t_3 * t_3) + (t_0 * t_0);
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = (t_1 * t_1) + (t_6 * t_6);
	float tmp;
	if (t_5 >= t_7) {
		tmp = (1.0f / sqrtf(fmaxf(t_5, t_7))) * t_3;
	} else {
		tmp = (powf(fmaxf((((t_2 * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_4)) * dX_46_u)), (((t_2 * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_4)) * dY_46_u))), -0.5f) * dY_46_u) * floorf(w);
	}
	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 = floor(h) ^ Float32(2.0)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6))
	tmp = Float32(0.0)
	if (t_5 >= t_7)
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_5, t_7))) * t_3);
	else
		tmp = Float32(Float32((fmax(Float32(Float32(Float32(t_2 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_4)) * dX_46_u)), Float32(Float32(Float32(t_2 * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_4)) * dY_46_u))) ^ Float32(-0.5)) * dY_46_u) * floor(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) ^ single(2.0);
	t_3 = floor(w) * dX_46_u;
	t_4 = floor(w) ^ single(2.0);
	t_5 = (t_3 * t_3) + (t_0 * t_0);
	t_6 = floor(h) * dY_46_v;
	t_7 = (t_1 * t_1) + (t_6 * t_6);
	tmp = single(0.0);
	if (t_5 >= t_7)
		tmp = (single(1.0) / sqrt(max(t_5, t_7))) * t_3;
	else
		tmp = ((max((((t_2 * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_4)) * dX_46_u)), (((t_2 * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_4)) * dY_46_u))) ^ single(-0.5)) * dY_46_u) * floor(w);
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

5. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]
6. Add Preprocessing

Alternative 3: 75.3% accurate, N/A× speedup

Math

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

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 (pow (floor h) 2.0))
        (t_3 (* (floor w) dX.u))
        (t_4 (pow (floor w) 2.0))
        (t_5 (+ (* t_3 t_3) (* t_0 t_0)))
        (t_6 (* (floor h) dY.v))
        (t_7 (+ (* t_1 t_1) (* t_6 t_6))))
   (if (>= t_5 t_7)
     (* (/ 1.0 (sqrt (fmax t_5 t_7))) t_3)
     (*
      (*
       (floor w)
       (pow
        (fmax
         (- (* (* t_2 dX.v) dX.v) (* (* -1.0 (* dX.u t_4)) dX.u))
         (- (* (* t_2 dY.v) dY.v) (* (* -1.0 (* dY.u t_4)) dY.u)))
        -0.5))
      dY.u))))

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 = powf(floorf(h), 2.0f);
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = (t_3 * t_3) + (t_0 * t_0);
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = (t_1 * t_1) + (t_6 * t_6);
	float tmp;
	if (t_5 >= t_7) {
		tmp = (1.0f / sqrtf(fmaxf(t_5, t_7))) * t_3;
	} else {
		tmp = (floorf(w) * powf(fmaxf((((t_2 * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_4)) * dX_46_u)), (((t_2 * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_4)) * dY_46_u))), -0.5f)) * dY_46_u;
	}
	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 = floor(h) ^ Float32(2.0)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6))
	tmp = Float32(0.0)
	if (t_5 >= t_7)
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_5, t_7))) * t_3);
	else
		tmp = Float32(Float32(floor(w) * (fmax(Float32(Float32(Float32(t_2 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_4)) * dX_46_u)), Float32(Float32(Float32(t_2 * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_4)) * dY_46_u))) ^ Float32(-0.5))) * dY_46_u);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) ^ single(2.0);
	t_3 = floor(w) * dX_46_u;
	t_4 = floor(w) ^ single(2.0);
	t_5 = (t_3 * t_3) + (t_0 * t_0);
	t_6 = floor(h) * dY_46_v;
	t_7 = (t_1 * t_1) + (t_6 * t_6);
	tmp = single(0.0);
	if (t_5 >= t_7)
		tmp = (single(1.0) / sqrt(max(t_5, t_7))) * t_3;
	else
		tmp = (floor(w) * (max((((t_2 * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_4)) * dX_46_u)), (((t_2 * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_4)) * dY_46_u))) ^ single(-0.5))) * dY_46_u;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dY.u\\ \end{array} \]

5. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dY.u\\ \end{array} \]
6. Add Preprocessing

Alternative 4: 75.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := t\_1 \cdot dY.v\\ t_5 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_6 := \left\lfloor w\right\rfloor \cdot dX.u\\ \mathbf{if}\;t\_6 \cdot t\_6 + t\_0 \cdot t\_0 \geq t\_3 \cdot t\_3 + t\_5 \cdot t\_5:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, t\_1, \left(dX.u \cdot dX.u\right) \cdot t\_2\right), \mathsf{fma}\left(t\_4, dY.v, \left(dY.u \cdot dY.u\right) \cdot t\_2\right)\right)\right)}^{-0.5} \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left(t\_1 \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot t\_2\right)\right) \cdot dX.u, t\_4 \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot t\_2\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dY.u\\ \end{array} \end{array} \]

FPCore

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

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 = powf(floorf(h), 2.0f);
	float t_2 = powf(floorf(w), 2.0f);
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = t_1 * dY_46_v;
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = floorf(w) * dX_46_u;
	float tmp;
	if (((t_6 * t_6) + (t_0 * t_0)) >= ((t_3 * t_3) + (t_5 * t_5))) {
		tmp = (powf(fmaxf(fmaf((dX_46_v * dX_46_v), t_1, ((dX_46_u * dX_46_u) * t_2)), fmaf(t_4, dY_46_v, ((dY_46_u * dY_46_u) * t_2))), -0.5f) * floorf(w)) * dX_46_u;
	} else {
		tmp = (floorf(w) * powf(fmaxf((((t_1 * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_2)) * dX_46_u)), ((t_4 * dY_46_v) - ((-1.0f * (dY_46_u * t_2)) * dY_46_u))), -0.5f)) * dY_46_u;
	}
	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 = floor(h) ^ Float32(2.0)
	t_2 = floor(w) ^ Float32(2.0)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(t_1 * dY_46_v)
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (Float32(Float32(t_6 * t_6) + Float32(t_0 * t_0)) >= Float32(Float32(t_3 * t_3) + Float32(t_5 * t_5)))
		tmp = Float32(Float32((fmax(fma(Float32(dX_46_v * dX_46_v), t_1, Float32(Float32(dX_46_u * dX_46_u) * t_2)), fma(t_4, dY_46_v, Float32(Float32(dY_46_u * dY_46_u) * t_2))) ^ Float32(-0.5)) * floor(w)) * dX_46_u);
	else
		tmp = Float32(Float32(floor(w) * (fmax(Float32(Float32(Float32(t_1 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_2)) * dX_46_u)), Float32(Float32(t_4 * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_2)) * dY_46_u))) ^ Float32(-0.5))) * dY_46_u);
	end
	return tmp
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dY.u\\ \end{array} \]

6. Applied rewrites 76.7%

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

7. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left({\left(\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}^{-0.5} \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dY.u\\ \end{array} \]
8. Add Preprocessing

Alternative 5: 75.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(Float32(t_4 * t_4) + Float32(t_0 * t_0))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_2 * t_2) + Float32(t_6 * t_6))
	tmp = Float32(0.0)
	if (t_5 >= t_7)
		tmp = Float32(Float32(floor(w) * (fmax(Float32(Float32(Float32(t_1 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_3)) * dX_46_u)), Float32(Float32(Float32(t_1 * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_3)) * dY_46_u))) ^ Float32(-0.5))) * dX_46_u);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_5, t_7))) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(h) ^ single(2.0);
	t_2 = floor(w) * dY_46_u;
	t_3 = floor(w) ^ single(2.0);
	t_4 = floor(w) * dX_46_u;
	t_5 = (t_4 * t_4) + (t_0 * t_0);
	t_6 = floor(h) * dY_46_v;
	t_7 = (t_2 * t_2) + (t_6 * t_6);
	tmp = single(0.0);
	if (t_5 >= t_7)
		tmp = (floor(w) * (max((((t_1 * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_3)) * dX_46_u)), (((t_1 * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_3)) * dY_46_u))) ^ single(-0.5))) * dX_46_u;
	else
		tmp = (single(1.0) / sqrt(max(t_5, t_7))) * t_2;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.7%

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

5. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
6. Add Preprocessing

Alternative 6: 75.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_4 := \log \left(\left\lfloor h\right\rfloor \right) \cdot 2\\ t_5 := t\_2 \cdot t\_2 + t\_0 \cdot t\_0\\ t_6 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_7 := t\_1 \cdot t\_1 + t\_6 \cdot t\_6\\ \mathbf{if}\;t\_5 \geq t\_7:\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh t\_4 + \sinh t\_4\right) \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot t\_3\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot t\_3\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_5, t\_7\right)}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) dX.u))
        (t_3 (pow (floor w) 2.0))
        (t_4 (* (log (floor h)) 2.0))
        (t_5 (+ (* t_2 t_2) (* t_0 t_0)))
        (t_6 (* (floor h) dY.v))
        (t_7 (+ (* t_1 t_1) (* t_6 t_6))))
   (if (>= t_5 t_7)
     (*
      (*
       (floor w)
       (pow
        (fmax
         (-
          (* (* (+ (cosh t_4) (sinh t_4)) dX.v) dX.v)
          (* (* -1.0 (* dX.u t_3)) dX.u))
         (-
          (* (* (pow (floor h) 2.0) dY.v) dY.v)
          (* (* -1.0 (* dY.u t_3)) dY.u)))
        -0.5))
      dX.u)
     (* (/ 1.0 (sqrt (fmax t_5 t_7))) t_1))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = logf(floorf(h)) * 2.0f;
	float t_5 = (t_2 * t_2) + (t_0 * t_0);
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = (t_1 * t_1) + (t_6 * t_6);
	float tmp;
	if (t_5 >= t_7) {
		tmp = (floorf(w) * powf(fmaxf(((((coshf(t_4) + sinhf(t_4)) * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_3)) * dX_46_u)), (((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_3)) * dY_46_u))), -0.5f)) * dX_46_u;
	} else {
		tmp = (1.0f / sqrtf(fmaxf(t_5, t_7))) * t_1;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = Float32(log(floor(h)) * Float32(2.0))
	t_5 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(Float32(t_1 * t_1) + Float32(t_6 * t_6))
	tmp = Float32(0.0)
	if (t_5 >= t_7)
		tmp = Float32(Float32(floor(w) * (fmax(Float32(Float32(Float32(Float32(cosh(t_4) + sinh(t_4)) * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_3)) * dX_46_u)), Float32(Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_3)) * dY_46_u))) ^ Float32(-0.5))) * dX_46_u);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_5, t_7))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) * dX_46_v;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(w) * dX_46_u;
	t_3 = floor(w) ^ single(2.0);
	t_4 = log(floor(h)) * single(2.0);
	t_5 = (t_2 * t_2) + (t_0 * t_0);
	t_6 = floor(h) * dY_46_v;
	t_7 = (t_1 * t_1) + (t_6 * t_6);
	tmp = single(0.0);
	if (t_5 >= t_7)
		tmp = (floor(w) * (max(((((cosh(t_4) + sinh(t_4)) * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_3)) * dX_46_u)), ((((floor(h) ^ single(2.0)) * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_3)) * dY_46_u))) ^ single(-0.5))) * dX_46_u;
	else
		tmp = (single(1.0) / sqrt(max(t_5, t_7))) * t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.u}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

   1. lift-floor.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. pow-to-exp N/A

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

   4. sinh-+-cosh-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lower-cosh.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. lift-floor.f32 N/A

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

   10. lower-sinh.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lift-floor.f32 76.3

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \color{blue}{\left(\left\lfloor h\right\rfloor \right)} \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\color{blue}{\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right)} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5}\right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\left(\left\lfloor w\right\rfloor \cdot {\left(\mathsf{max}\left(\left(\left(\cosh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right) + \sinh \left(\log \left(\left\lfloor h\right\rfloor \right) \cdot 2\right)\right) \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5}\right) \cdot dX.u\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
8. Add Preprocessing

Alternative 7: 59.6% accurate, N/A× speedup

Math

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

FPCore

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

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 = powf(floorf(h), 2.0f);
	float t_1 = t_0 * dY_46_v;
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(floorf(w), 2.0f);
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = floorf(h) * dX_46_v;
	float t_7 = (t_0 * dX_46_v) * dX_46_v;
	float tmp;
	if (((t_5 * t_5) + (t_6 * t_6)) >= ((t_2 * t_2) + (t_4 * t_4))) {
		tmp = sqrtf((1.0f / fmaxf(t_7, fmaf(t_1, dY_46_v, ((dY_46_u * dY_46_u) * t_3))))) * t_5;
	} else {
		tmp = (powf(fmaxf((t_7 - ((-1.0f * (dX_46_u * t_3)) * dX_46_u)), ((t_1 * dY_46_v) - ((-1.0f * (dY_46_u * t_3)) * dY_46_u))), -0.5f) * dY_46_u) * floorf(w);
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(t_0 * dY_46_v)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = floor(w) ^ Float32(2.0)
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = Float32(floor(h) * dX_46_v)
	t_7 = Float32(Float32(t_0 * dX_46_v) * dX_46_v)
	tmp = Float32(0.0)
	if (Float32(Float32(t_5 * t_5) + Float32(t_6 * t_6)) >= Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)))
		tmp = Float32(sqrt(Float32(Float32(1.0) / fmax(t_7, fma(t_1, dY_46_v, Float32(Float32(dY_46_u * dY_46_u) * t_3))))) * t_5);
	else
		tmp = Float32(Float32((fmax(Float32(t_7 - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_3)) * dX_46_u)), Float32(Float32(t_1 * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_3)) * dY_46_u))) ^ Float32(-0.5)) * dY_46_u) * floor(w));
	end
	return tmp
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

5. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. lift-*.f32 63.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

7. Applied rewrites 63.0%

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

9. Applied rewrites 63.0%

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

10. Final simplification 63.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]
11. Add Preprocessing

Alternative 8: 58.4% accurate, N/A× speedup

Math

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

FPCore

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

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 = powf(floorf(h), 2.0f);
	float t_2 = (t_1 * dX_46_v) * dX_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = floorf(w) * dY_46_u;
	float t_5 = t_1 * dY_46_v;
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = powf(floorf(w), 2.0f);
	float tmp;
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_4 * t_4) + (t_6 * t_6))) {
		tmp = expf((logf(sqrtf(fmaxf(t_2, fmaf(t_5, dY_46_v, ((dY_46_u * dY_46_u) * t_7))))) * -1.0f)) * ((1.0f / powf(floorf(w), -1.0f)) * dX_46_u);
	} else {
		tmp = (powf(fmaxf((t_2 - ((-1.0f * (dX_46_u * t_7)) * dX_46_u)), ((t_5 * dY_46_v) - ((-1.0f * (dY_46_u * t_7)) * dY_46_u))), -0.5f) * dY_46_u) * floorf(w);
	}
	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 = floor(h) ^ Float32(2.0)
	t_2 = Float32(Float32(t_1 * dX_46_v) * dX_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = Float32(floor(w) * dY_46_u)
	t_5 = Float32(t_1 * dY_46_v)
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) >= Float32(Float32(t_4 * t_4) + Float32(t_6 * t_6)))
		tmp = Float32(exp(Float32(log(sqrt(fmax(t_2, fma(t_5, dY_46_v, Float32(Float32(dY_46_u * dY_46_u) * t_7))))) * Float32(-1.0))) * Float32(Float32(Float32(1.0) / (floor(w) ^ Float32(-1.0))) * dX_46_u));
	else
		tmp = Float32(Float32((fmax(Float32(t_2 - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_7)) * dX_46_u)), Float32(Float32(t_5 * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_7)) * dY_46_u))) ^ Float32(-0.5)) * dY_46_u) * floor(w));
	end
	return tmp
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

5. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. lift-*.f32 63.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

7. Applied rewrites 63.0%

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

9. Applied rewrites 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]
10. Step-by-step derivation

    1. lift-floor.f32 N/A

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

    2. unpow1 N/A

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

    3. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1} \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}} \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{\frac{-1}{2}} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

    4. pow-neg N/A

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

    5. lower-/.f32 N/A

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

    6. lower-pow.f32 N/A

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

    7. lift-floor.f32 61.5

       \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1} \cdot \left(\frac{1}{{\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{-1}} \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

11. Applied rewrites 61.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1} \cdot \left(\color{blue}{\frac{1}{{\left(\left\lfloor w\right\rfloor \right)}^{-1}}} \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

12. Final simplification 61.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1} \cdot \left(\frac{1}{{\left(\left\lfloor w\right\rfloor \right)}^{-1}} \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]
13. Add Preprocessing

Alternative 9: 58.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := \left(t\_1 \cdot dX.v\right) \cdot dX.v\\ t_3 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_4 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_5 := t\_1 \cdot dY.v\\ t_6 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_7 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;t\_3 \cdot t\_3 + t\_0 \cdot t\_0 \geq t\_4 \cdot t\_4 + t\_6 \cdot t\_6:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(t\_2, \mathsf{fma}\left(t\_5, dY.v, \left(dY.u \cdot dY.u\right) \cdot t\_7\right)\right)}\right) \cdot -1} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(t\_2 - \left(-1 \cdot \left(dX.u \cdot t\_7\right)\right) \cdot dX.u, t\_5 \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot t\_7\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \end{array} \]

FPCore

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

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 = powf(floorf(h), 2.0f);
	float t_2 = (t_1 * dX_46_v) * dX_46_v;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = floorf(w) * dY_46_u;
	float t_5 = t_1 * dY_46_v;
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = powf(floorf(w), 2.0f);
	float tmp;
	if (((t_3 * t_3) + (t_0 * t_0)) >= ((t_4 * t_4) + (t_6 * t_6))) {
		tmp = expf((logf(sqrtf(fmaxf(t_2, fmaf(t_5, dY_46_v, ((dY_46_u * dY_46_u) * t_7))))) * -1.0f)) * t_3;
	} else {
		tmp = (powf(fmaxf((t_2 - ((-1.0f * (dX_46_u * t_7)) * dX_46_u)), ((t_5 * dY_46_v) - ((-1.0f * (dY_46_u * t_7)) * dY_46_u))), -0.5f) * dY_46_u) * floorf(w);
	}
	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 = floor(h) ^ Float32(2.0)
	t_2 = Float32(Float32(t_1 * dX_46_v) * dX_46_v)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = Float32(floor(w) * dY_46_u)
	t_5 = Float32(t_1 * dY_46_v)
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) >= Float32(Float32(t_4 * t_4) + Float32(t_6 * t_6)))
		tmp = Float32(exp(Float32(log(sqrt(fmax(t_2, fma(t_5, dY_46_v, Float32(Float32(dY_46_u * dY_46_u) * t_7))))) * Float32(-1.0))) * t_3);
	else
		tmp = Float32(Float32((fmax(Float32(t_2 - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_7)) * dX_46_u)), Float32(Float32(t_5 * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_7)) * dY_46_u))) ^ Float32(-0.5)) * dY_46_u) * floor(w));
	end
	return tmp
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

5. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. lift-*.f32 63.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

7. Applied rewrites 63.0%

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

9. Applied rewrites 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

10. Final simplification 61.4%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]
11. Add Preprocessing

Alternative 10: 58.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_1 := \left(t\_0 \cdot dX.v\right) \cdot dX.v\\ t_2 := \left(t\_0 \cdot dY.v\right) \cdot dY.v\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_7 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;t\_5 \cdot t\_5 + t\_6 \cdot t\_6 \geq t\_3 \cdot t\_3 + t\_4 \cdot t\_4:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(t\_1, t\_2\right)}\right) \cdot -1} \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(t\_1 - \left(-1 \cdot \left(dX.u \cdot t\_7\right)\right) \cdot dX.u, t\_2 - \left(-1 \cdot \left(dY.u \cdot t\_7\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0))
        (t_1 (* (* t_0 dX.v) dX.v))
        (t_2 (* (* t_0 dY.v) dY.v))
        (t_3 (* (floor w) dY.u))
        (t_4 (* (floor h) dY.v))
        (t_5 (* (floor w) dX.u))
        (t_6 (* (floor h) dX.v))
        (t_7 (pow (floor w) 2.0)))
   (if (>= (+ (* t_5 t_5) (* t_6 t_6)) (+ (* t_3 t_3) (* t_4 t_4)))
     (* (exp (* (log (sqrt (fmax t_1 t_2))) -1.0)) t_5)
     (*
      (*
       (pow
        (fmax
         (- t_1 (* (* -1.0 (* dX.u t_7)) dX.u))
         (- t_2 (* (* -1.0 (* dY.u t_7)) dY.u)))
        -0.5)
       dY.u)
      (floor w)))))

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 = powf(floorf(h), 2.0f);
	float t_1 = (t_0 * dX_46_v) * dX_46_v;
	float t_2 = (t_0 * dY_46_v) * dY_46_v;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = floorf(h) * dX_46_v;
	float t_7 = powf(floorf(w), 2.0f);
	float tmp;
	if (((t_5 * t_5) + (t_6 * t_6)) >= ((t_3 * t_3) + (t_4 * t_4))) {
		tmp = expf((logf(sqrtf(fmaxf(t_1, t_2))) * -1.0f)) * t_5;
	} else {
		tmp = (powf(fmaxf((t_1 - ((-1.0f * (dX_46_u * t_7)) * dX_46_u)), (t_2 - ((-1.0f * (dY_46_u * t_7)) * dY_46_u))), -0.5f) * dY_46_u) * floorf(w);
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = floor(h) ^ Float32(2.0)
	t_1 = Float32(Float32(t_0 * dX_46_v) * dX_46_v)
	t_2 = Float32(Float32(t_0 * dY_46_v) * dY_46_v)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = Float32(floor(h) * dX_46_v)
	t_7 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_5 * t_5) + Float32(t_6 * t_6)) >= Float32(Float32(t_3 * t_3) + Float32(t_4 * t_4)))
		tmp = Float32(exp(Float32(log(sqrt(fmax(t_1, t_2))) * Float32(-1.0))) * t_5);
	else
		tmp = Float32(Float32((fmax(Float32(t_1 - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_7)) * dX_46_u)), Float32(t_2 - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_7)) * dY_46_u))) ^ Float32(-0.5)) * dY_46_u) * floor(w));
	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) ^ single(2.0);
	t_1 = (t_0 * dX_46_v) * dX_46_v;
	t_2 = (t_0 * dY_46_v) * dY_46_v;
	t_3 = floor(w) * dY_46_u;
	t_4 = floor(h) * dY_46_v;
	t_5 = floor(w) * dX_46_u;
	t_6 = floor(h) * dX_46_v;
	t_7 = floor(w) ^ single(2.0);
	tmp = single(0.0);
	if (((t_5 * t_5) + (t_6 * t_6)) >= ((t_3 * t_3) + (t_4 * t_4)))
		tmp = exp((log(sqrt(max(t_1, t_2))) * single(-1.0))) * t_5;
	else
		tmp = ((max((t_1 - ((single(-1.0) * (dX_46_u * t_7)) * dX_46_u)), (t_2 - ((single(-1.0) * (dY_46_u * t_7)) * dY_46_u))) ^ single(-0.5)) * dY_46_u) * floor(w);
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 76.8%

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

4. Applied rewrites 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right), \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

5. Taylor expanded in dX.u around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. lift-*.f32 63.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

7. Applied rewrites 63.0%

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

9. Applied rewrites 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\color{blue}{e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.v, \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right) \cdot -1}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

10. Taylor expanded in dY.u around 0

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

12. Applied rewrites 61.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}\right)}\right) \cdot -1} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dX.u\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot \left(-dY.u\right)\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]

13. Final simplification 61.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right) \cdot -1} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dX.u, \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \\ \end{array} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dX.u)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dY.u))))