Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 76.3% → 76.2%

Time: 28.0s

Alternatives: 7

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: 76.3% 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: 76.2% accurate, N/A× speedup

Math

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

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

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 78.1%

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

6. Applied rewrites 78.1%

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

7. Final simplification 78.1%

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

Alternative 2: 76.2% accurate, N/A× speedup

Math

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

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 t_2 1.0))
        (t_4 (* -1.0 t_2))
        (t_5 (+ (* t_4 t_4) (* 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 (fma t_3 t_3 (* (* (pow (floor h) 2.0) dX.v) dX.v)) t_7)))
      t_2)
     (* (/ 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(t_2, 1.0f);
	float t_4 = -1.0f * t_2;
	float t_5 = (t_4 * t_4) + (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(fmaf(t_3, t_3, ((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v)), t_7))) * t_2;
	} 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 = t_2 ^ Float32(1.0)
	t_4 = Float32(Float32(-1.0) * t_2)
	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_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(fma(t_3, t_3, Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)), t_7))) * t_2);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_5, t_7))) * t_1);
	end
	return tmp
end

Derivation

1. Initial program 78.0%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. unpow-prod-up N/A

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

   6. lift-*.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-floor.f32 N/A

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

   10. *-commutative N/A

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

   11. pow-prod-down N/A

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

   12. lower-fma.f32 N/A

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

4. Applied rewrites 78.1%

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

5. Final simplification 78.1%

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

Alternative 3: 76.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_5 := -1 \cdot t\_2\\ t_6 := t\_5 \cdot t\_5 + t\_0 \cdot t\_0\\ t_7 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_8 := t\_3 \cdot t\_3 + t\_7 \cdot t\_7\\ \mathbf{if}\;t\_6 \geq t\_8:\\ \;\;\;\;{\left(\mathsf{max}\left(\left(t\_1 \cdot dX.v\right) \cdot dX.v - \left(-1 \cdot \left(dX.u \cdot t\_4\right)\right) \cdot dX.u, \left(t\_1 \cdot dY.v\right) \cdot dY.v - \left(-1 \cdot \left(dY.u \cdot t\_4\right)\right) \cdot dY.u\right)\right)}^{-0.5} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{max}\left(t\_6, t\_8\right)}} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* (floor w) dX.u))
        (t_3 (* (floor w) dY.u))
        (t_4 (pow (floor w) 2.0))
        (t_5 (* -1.0 t_2))
        (t_6 (+ (* t_5 t_5) (* t_0 t_0)))
        (t_7 (* (floor h) dY.v))
        (t_8 (+ (* t_3 t_3) (* t_7 t_7))))
   (if (>= t_6 t_8)
     (*
      (pow
       (fmax
        (- (* (* t_1 dX.v) dX.v) (* (* -1.0 (* dX.u t_4)) dX.u))
        (- (* (* t_1 dY.v) dY.v) (* (* -1.0 (* dY.u t_4)) dY.u)))
       -0.5)
      t_2)
     (* (/ 1.0 (sqrt (fmax t_6 t_8))) t_3))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = floorf(w) * dX_46_u;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = powf(floorf(w), 2.0f);
	float t_5 = -1.0f * t_2;
	float t_6 = (t_5 * t_5) + (t_0 * t_0);
	float t_7 = floorf(h) * dY_46_v;
	float t_8 = (t_3 * t_3) + (t_7 * t_7);
	float tmp;
	if (t_6 >= t_8) {
		tmp = powf(fmaxf((((t_1 * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_4)) * dX_46_u)), (((t_1 * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_4)) * dY_46_u))), -0.5f) * t_2;
	} else {
		tmp = (1.0f / sqrtf(fmaxf(t_6, t_8))) * t_3;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = floor(w) ^ Float32(2.0)
	t_5 = Float32(Float32(-1.0) * t_2)
	t_6 = Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0))
	t_7 = Float32(floor(h) * dY_46_v)
	t_8 = Float32(Float32(t_3 * t_3) + Float32(t_7 * t_7))
	tmp = Float32(0.0)
	if (t_6 >= t_8)
		tmp = Float32((fmax(Float32(Float32(Float32(t_1 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_4)) * dX_46_u)), Float32(Float32(Float32(t_1 * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_4)) * dY_46_u))) ^ Float32(-0.5)) * t_2);
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_6, t_8))) * t_3);
	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) * dX_46_u;
	t_3 = floor(w) * dY_46_u;
	t_4 = floor(w) ^ single(2.0);
	t_5 = single(-1.0) * t_2;
	t_6 = (t_5 * t_5) + (t_0 * t_0);
	t_7 = floor(h) * dY_46_v;
	t_8 = (t_3 * t_3) + (t_7 * t_7);
	tmp = single(0.0);
	if (t_6 >= t_8)
		tmp = (max((((t_1 * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_4)) * dX_46_u)), (((t_1 * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_4)) * dY_46_u))) ^ single(-0.5)) * t_2;
	else
		tmp = (single(1.0) / sqrt(max(t_6, t_8))) * t_3;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in w around 0

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

5. Applied rewrites 78.0%

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

6. Final simplification 78.0%

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

Alternative 4: 49.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := \mathsf{fma}\left(t\_1 \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_3 := \left(-1 \cdot dY.v\right) \cdot dY.v\\ t_4 := \left(dY.u \cdot dY.u\right) \cdot t\_0 - t\_3 \cdot t\_1\\ \mathbf{if}\;t\_2 \geq t\_4:\\ \;\;\;\;{\left(\mathsf{max}\left(t\_2, t\_4\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(t\_2, \frac{{t\_0}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {t\_3}^{3} \cdot {t\_1}^{3}}{\mathsf{fma}\left(\left(-1 \cdot dY.u\right) \cdot dY.u, \left(t\_0 \cdot t\_1\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({t\_1}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {t\_0}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (pow (floor h) 2.0))
        (t_2 (fma (* t_1 dX.v) dX.v (* (* dX.u dX.u) t_0)))
        (t_3 (* (* -1.0 dY.v) dY.v))
        (t_4 (- (* (* dY.u dY.u) t_0) (* t_3 t_1))))
   (if (>= t_2 t_4)
     (* (pow (fmax t_2 t_4) -0.5) (* (floor w) dX.u))
     (*
      (pow
       (fmax
        t_2
        (/
         (-
          (* (pow t_0 3.0) (pow (* (* dY.u dY.u) dY.u) 2.0))
          (* (pow t_3 3.0) (pow t_1 3.0)))
         (fma
          (* (* -1.0 dY.u) dY.u)
          (* (* t_0 t_1) (* dY.v dY.v))
          (fma
           (pow t_1 2.0)
           (pow (* dY.v dY.v) 2.0)
           (* (pow t_0 2.0) (pow (* dY.u dY.u) 2.0))))))
       -0.5)
      (* (floor w) 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 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = fmaf((t_1 * dX_46_v), dX_46_v, ((dX_46_u * dX_46_u) * t_0));
	float t_3 = (-1.0f * dY_46_v) * dY_46_v;
	float t_4 = ((dY_46_u * dY_46_u) * t_0) - (t_3 * t_1);
	float tmp;
	if (t_2 >= t_4) {
		tmp = powf(fmaxf(t_2, t_4), -0.5f) * (floorf(w) * dX_46_u);
	} else {
		tmp = powf(fmaxf(t_2, (((powf(t_0, 3.0f) * powf(((dY_46_u * dY_46_u) * dY_46_u), 2.0f)) - (powf(t_3, 3.0f) * powf(t_1, 3.0f))) / fmaf(((-1.0f * dY_46_u) * dY_46_u), ((t_0 * t_1) * (dY_46_v * dY_46_v)), fmaf(powf(t_1, 2.0f), powf((dY_46_v * dY_46_v), 2.0f), (powf(t_0, 2.0f) * powf((dY_46_u * dY_46_u), 2.0f)))))), -0.5f) * (floorf(w) * 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 = floor(w) ^ Float32(2.0)
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = fma(Float32(t_1 * dX_46_v), dX_46_v, Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_3 = Float32(Float32(Float32(-1.0) * dY_46_v) * dY_46_v)
	t_4 = Float32(Float32(Float32(dY_46_u * dY_46_u) * t_0) - Float32(t_3 * t_1))
	tmp = Float32(0.0)
	if (t_2 >= t_4)
		tmp = Float32((fmax(t_2, t_4) ^ Float32(-0.5)) * Float32(floor(w) * dX_46_u));
	else
		tmp = Float32((fmax(t_2, Float32(Float32(Float32((t_0 ^ Float32(3.0)) * (Float32(Float32(dY_46_u * dY_46_u) * dY_46_u) ^ Float32(2.0))) - Float32((t_3 ^ Float32(3.0)) * (t_1 ^ Float32(3.0)))) / fma(Float32(Float32(Float32(-1.0) * dY_46_u) * dY_46_u), Float32(Float32(t_0 * t_1) * Float32(dY_46_v * dY_46_v)), fma((t_1 ^ Float32(2.0)), (Float32(dY_46_v * dY_46_v) ^ Float32(2.0)), Float32((t_0 ^ Float32(2.0)) * (Float32(dY_46_u * dY_46_u) ^ Float32(2.0))))))) ^ Float32(-0.5)) * Float32(floor(w) * dY_46_u));
	end
	return tmp
end

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 51.3%

   \[\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):\\ \;\;\;\;\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

   1. lift-*.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(\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. pow2 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. lift-*.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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow-prod-down 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(\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 \right)}^{2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   6. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot e^{\log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. prod-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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   9. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. lower-fma.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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   12. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. lower-log.f32 48.7

       \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 48.7%

   \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left(e^{2 \cdot \log dX.v + 2 \cdot \log \left(\left\lfloor h\right\rfloor \right)} + {dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \frac{{dY.u}^{6} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{6} - -1 \cdot \left({dY.v}^{6} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{6}\right)}{-1 \cdot \left({dY.u}^{2} \cdot \left({dY.v}^{2} \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right) + \left({dY.u}^{4} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{4} + {dY.v}^{4} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{4}\right)}\right)}}\\ } \end{array}} \]

9. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

10. Final simplification 51.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(\left(-1 \cdot dY.u\right) \cdot dY.u, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
11. Add Preprocessing

Alternative 5: 49.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_2 := t\_1 \cdot dX.v\\ t_3 := \mathsf{fma}\left(t\_2, dX.v, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_4 := \left(-1 \cdot dY.v\right) \cdot dY.v\\ t_5 := \left(dY.u \cdot dY.u\right) \cdot t\_0 - t\_4 \cdot t\_1\\ \mathbf{if}\;t\_3 \geq t\_5:\\ \;\;\;\;{\left(\mathsf{max}\left(t\_3, t\_5\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left(t\_2, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{t\_0}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {t\_4}^{3} \cdot {t\_1}^{3}}{\mathsf{fma}\left(\left(-1 \cdot dY.u\right) \cdot dY.u, \left(t\_0 \cdot t\_1\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({t\_1}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {t\_0}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (pow (floor h) 2.0))
        (t_2 (* t_1 dX.v))
        (t_3 (fma t_2 dX.v (* (* dX.u dX.u) t_0)))
        (t_4 (* (* -1.0 dY.v) dY.v))
        (t_5 (- (* (* dY.u dY.u) t_0) (* t_4 t_1))))
   (if (>= t_3 t_5)
     (* (pow (fmax t_3 t_5) -0.5) (* (floor w) dX.u))
     (*
      (pow
       (fmax
        (fma t_2 dX.v (* (* dX.u dX.u) (pow (exp (log (floor w))) 2.0)))
        (/
         (-
          (* (pow t_0 3.0) (pow (* (* dY.u dY.u) dY.u) 2.0))
          (* (pow t_4 3.0) (pow t_1 3.0)))
         (fma
          (* (* -1.0 dY.u) dY.u)
          (* (* t_0 t_1) (* dY.v dY.v))
          (fma
           (pow t_1 2.0)
           (pow (* dY.v dY.v) 2.0)
           (* (pow t_0 2.0) (pow (* dY.u dY.u) 2.0))))))
       -0.5)
      (* (floor w) 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 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(h), 2.0f);
	float t_2 = t_1 * dX_46_v;
	float t_3 = fmaf(t_2, dX_46_v, ((dX_46_u * dX_46_u) * t_0));
	float t_4 = (-1.0f * dY_46_v) * dY_46_v;
	float t_5 = ((dY_46_u * dY_46_u) * t_0) - (t_4 * t_1);
	float tmp;
	if (t_3 >= t_5) {
		tmp = powf(fmaxf(t_3, t_5), -0.5f) * (floorf(w) * dX_46_u);
	} else {
		tmp = powf(fmaxf(fmaf(t_2, dX_46_v, ((dX_46_u * dX_46_u) * powf(expf(logf(floorf(w))), 2.0f))), (((powf(t_0, 3.0f) * powf(((dY_46_u * dY_46_u) * dY_46_u), 2.0f)) - (powf(t_4, 3.0f) * powf(t_1, 3.0f))) / fmaf(((-1.0f * dY_46_u) * dY_46_u), ((t_0 * t_1) * (dY_46_v * dY_46_v)), fmaf(powf(t_1, 2.0f), powf((dY_46_v * dY_46_v), 2.0f), (powf(t_0, 2.0f) * powf((dY_46_u * dY_46_u), 2.0f)))))), -0.5f) * (floorf(w) * 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 = floor(w) ^ Float32(2.0)
	t_1 = floor(h) ^ Float32(2.0)
	t_2 = Float32(t_1 * dX_46_v)
	t_3 = fma(t_2, dX_46_v, Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_4 = Float32(Float32(Float32(-1.0) * dY_46_v) * dY_46_v)
	t_5 = Float32(Float32(Float32(dY_46_u * dY_46_u) * t_0) - Float32(t_4 * t_1))
	tmp = Float32(0.0)
	if (t_3 >= t_5)
		tmp = Float32((fmax(t_3, t_5) ^ Float32(-0.5)) * Float32(floor(w) * dX_46_u));
	else
		tmp = Float32((fmax(fma(t_2, dX_46_v, Float32(Float32(dX_46_u * dX_46_u) * (exp(log(floor(w))) ^ Float32(2.0)))), Float32(Float32(Float32((t_0 ^ Float32(3.0)) * (Float32(Float32(dY_46_u * dY_46_u) * dY_46_u) ^ Float32(2.0))) - Float32((t_4 ^ Float32(3.0)) * (t_1 ^ Float32(3.0)))) / fma(Float32(Float32(Float32(-1.0) * dY_46_u) * dY_46_u), Float32(Float32(t_0 * t_1) * Float32(dY_46_v * dY_46_v)), fma((t_1 ^ Float32(2.0)), (Float32(dY_46_v * dY_46_v) ^ Float32(2.0)), Float32((t_0 ^ Float32(2.0)) * (Float32(dY_46_u * dY_46_u) ^ Float32(2.0))))))) ^ Float32(-0.5)) * Float32(floor(w) * dY_46_u));
	end
	return tmp
end

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 51.3%

   \[\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):\\ \;\;\;\;\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

   1. lift-*.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(\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. pow2 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. lift-*.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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow-prod-down 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(\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 \right)}^{2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   6. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot e^{\log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. prod-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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   9. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. lower-fma.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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   12. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. lower-log.f32 48.7

       \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 48.7%

   \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left(e^{2 \cdot \log dX.v + 2 \cdot \log \left(\left\lfloor h\right\rfloor \right)} + {dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \frac{{dY.u}^{6} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{6} - -1 \cdot \left({dY.v}^{6} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{6}\right)}{-1 \cdot \left({dY.u}^{2} \cdot \left({dY.v}^{2} \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right) + \left({dY.u}^{4} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{4} + {dY.v}^{4} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{4}\right)}\right)}}\\ } \end{array}} \]

9. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]
10. Step-by-step derivation

    1. lift-floor.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    2. rem-exp-log N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    3. lower-exp.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    4. lift-log.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{\frac{-1}{2}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

    5. lift-floor.f32 51.3

       \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

11. Applied rewrites 51.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

12. Final simplification 51.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(e^{\log \left(\left\lfloor w\right\rfloor \right)}\right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(\left(-1 \cdot dY.u\right) \cdot dY.u, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
13. Add Preprocessing

Alternative 6: 49.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dY.u \cdot \left(t\_0 \cdot dY.u\right)\\ t_2 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := \left(dY.u \cdot dY.u\right) \cdot t\_0 - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot t\_2\\ t_4 := t\_2 \cdot dY.v\\ t_5 := t\_2 \cdot dX.v\\ t_6 := \mathsf{fma}\left(t\_5, dX.v, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_7 := -1 \cdot \left(t\_1 \cdot -1\right)\\ \mathbf{if}\;t\_6 \geq t\_3:\\ \;\;\;\;{\left(\mathsf{max}\left(t\_6, t\_3\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\log \left(\mathsf{max}\left(t\_5 \cdot dX.v - \left(\left(t\_0 \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{t\_1}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {t\_4}^{3}}{\mathsf{fma}\left(t\_7, t\_7, \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, t\_0 \cdot t\_2, {t\_4}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right)}\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dY.u (* t_0 dY.u)))
        (t_2 (pow (floor h) 2.0))
        (t_3 (- (* (* dY.u dY.u) t_0) (* (* (* -1.0 dY.v) dY.v) t_2)))
        (t_4 (* t_2 dY.v))
        (t_5 (* t_2 dX.v))
        (t_6 (fma t_5 dX.v (* (* dX.u dX.u) t_0)))
        (t_7 (* -1.0 (* t_1 -1.0))))
   (if (>= t_6 t_3)
     (* (pow (fmax t_6 t_3) -0.5) (* (floor w) dX.u))
     (*
      (pow
       (exp
        (log
         (fmax
          (- (* t_5 dX.v) (* (* (* t_0 dX.u) dX.u) -1.0))
          (/
           (- (pow t_1 3.0) (* (pow (* -1.0 dY.v) 3.0) (pow t_4 3.0)))
           (fma
            t_7
            t_7
            (*
             (fma (* (* dY.u dY.u) -1.0) (* t_0 t_2) (pow t_4 2.0))
             (* dY.v dY.v)))))))
       -0.5)
      (* (floor w) 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 = powf(floorf(w), 2.0f);
	float t_1 = dY_46_u * (t_0 * dY_46_u);
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = ((dY_46_u * dY_46_u) * t_0) - (((-1.0f * dY_46_v) * dY_46_v) * t_2);
	float t_4 = t_2 * dY_46_v;
	float t_5 = t_2 * dX_46_v;
	float t_6 = fmaf(t_5, dX_46_v, ((dX_46_u * dX_46_u) * t_0));
	float t_7 = -1.0f * (t_1 * -1.0f);
	float tmp;
	if (t_6 >= t_3) {
		tmp = powf(fmaxf(t_6, t_3), -0.5f) * (floorf(w) * dX_46_u);
	} else {
		tmp = powf(expf(logf(fmaxf(((t_5 * dX_46_v) - (((t_0 * dX_46_u) * dX_46_u) * -1.0f)), ((powf(t_1, 3.0f) - (powf((-1.0f * dY_46_v), 3.0f) * powf(t_4, 3.0f))) / fmaf(t_7, t_7, (fmaf(((dY_46_u * dY_46_u) * -1.0f), (t_0 * t_2), powf(t_4, 2.0f)) * (dY_46_v * dY_46_v))))))), -0.5f) * (floorf(w) * 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 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dY_46_u * Float32(t_0 * dY_46_u))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = Float32(Float32(Float32(dY_46_u * dY_46_u) * t_0) - Float32(Float32(Float32(Float32(-1.0) * dY_46_v) * dY_46_v) * t_2))
	t_4 = Float32(t_2 * dY_46_v)
	t_5 = Float32(t_2 * dX_46_v)
	t_6 = fma(t_5, dX_46_v, Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_7 = Float32(Float32(-1.0) * Float32(t_1 * Float32(-1.0)))
	tmp = Float32(0.0)
	if (t_6 >= t_3)
		tmp = Float32((fmax(t_6, t_3) ^ Float32(-0.5)) * Float32(floor(w) * dX_46_u));
	else
		tmp = Float32((exp(log(fmax(Float32(Float32(t_5 * dX_46_v) - Float32(Float32(Float32(t_0 * dX_46_u) * dX_46_u) * Float32(-1.0))), Float32(Float32((t_1 ^ Float32(3.0)) - Float32((Float32(Float32(-1.0) * dY_46_v) ^ Float32(3.0)) * (t_4 ^ Float32(3.0)))) / fma(t_7, t_7, Float32(fma(Float32(Float32(dY_46_u * dY_46_u) * Float32(-1.0)), Float32(t_0 * t_2), (t_4 ^ Float32(2.0))) * Float32(dY_46_v * dY_46_v))))))) ^ Float32(-0.5)) * Float32(floor(w) * dY_46_u));
	end
	return tmp
end

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 51.3%

   \[\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):\\ \;\;\;\;\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

   1. lift-*.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(\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. pow2 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. lift-*.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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow-prod-down 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(\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 \right)}^{2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   6. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot e^{\log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. prod-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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   9. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. lower-fma.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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   12. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. lower-log.f32 48.7

       \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 48.7%

   \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left(e^{2 \cdot \log dX.v + 2 \cdot \log \left(\left\lfloor h\right\rfloor \right)} + {dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \frac{{dY.u}^{6} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{6} - -1 \cdot \left({dY.v}^{6} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{6}\right)}{-1 \cdot \left({dY.u}^{2} \cdot \left({dY.v}^{2} \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right) + \left({dY.u}^{4} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{4} + {dY.v}^{4} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{4}\right)}\right)}}\\ } \end{array}} \]

9. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

11. Applied rewrites 51.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\log \left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{3}}{\mathsf{fma}\left(-\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1, -\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1, \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right)}\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

12. Final simplification 51.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\log \left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1\right), -1 \cdot \left(\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1\right), \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right)}\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
13. Add Preprocessing

Alternative 7: 44.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := dY.u \cdot \left(t\_0 \cdot dY.u\right)\\ t_2 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := \left(dY.u \cdot dY.u\right) \cdot t\_0 - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot t\_2\\ t_4 := t\_2 \cdot dY.v\\ t_5 := t\_2 \cdot dX.v\\ t_6 := \mathsf{fma}\left(t\_5, dX.v, \left(dX.u \cdot dX.u\right) \cdot t\_0\right)\\ t_7 := -1 \cdot \left(t\_1 \cdot -1\right)\\ t_8 := \log \left(\mathsf{max}\left(t\_5 \cdot dX.v - \left(\left(t\_0 \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{t\_1}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {t\_4}^{3}}{\mathsf{fma}\left(t\_7, t\_7, \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, t\_0 \cdot t\_2, {t\_4}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right) \cdot -0.5\\ \mathbf{if}\;t\_6 \geq t\_3:\\ \;\;\;\;{\left(\mathsf{max}\left(t\_6, t\_3\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh t\_8 + \sinh t\_8\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* dY.u (* t_0 dY.u)))
        (t_2 (pow (floor h) 2.0))
        (t_3 (- (* (* dY.u dY.u) t_0) (* (* (* -1.0 dY.v) dY.v) t_2)))
        (t_4 (* t_2 dY.v))
        (t_5 (* t_2 dX.v))
        (t_6 (fma t_5 dX.v (* (* dX.u dX.u) t_0)))
        (t_7 (* -1.0 (* t_1 -1.0)))
        (t_8
         (*
          (log
           (fmax
            (- (* t_5 dX.v) (* (* (* t_0 dX.u) dX.u) -1.0))
            (/
             (- (pow t_1 3.0) (* (pow (* -1.0 dY.v) 3.0) (pow t_4 3.0)))
             (fma
              t_7
              t_7
              (*
               (fma (* (* dY.u dY.u) -1.0) (* t_0 t_2) (pow t_4 2.0))
               (* dY.v dY.v))))))
          -0.5)))
   (if (>= t_6 t_3)
     (* (pow (fmax t_6 t_3) -0.5) (* (floor w) dX.u))
     (* (+ (cosh t_8) (sinh t_8)) (* (floor w) 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 = powf(floorf(w), 2.0f);
	float t_1 = dY_46_u * (t_0 * dY_46_u);
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = ((dY_46_u * dY_46_u) * t_0) - (((-1.0f * dY_46_v) * dY_46_v) * t_2);
	float t_4 = t_2 * dY_46_v;
	float t_5 = t_2 * dX_46_v;
	float t_6 = fmaf(t_5, dX_46_v, ((dX_46_u * dX_46_u) * t_0));
	float t_7 = -1.0f * (t_1 * -1.0f);
	float t_8 = logf(fmaxf(((t_5 * dX_46_v) - (((t_0 * dX_46_u) * dX_46_u) * -1.0f)), ((powf(t_1, 3.0f) - (powf((-1.0f * dY_46_v), 3.0f) * powf(t_4, 3.0f))) / fmaf(t_7, t_7, (fmaf(((dY_46_u * dY_46_u) * -1.0f), (t_0 * t_2), powf(t_4, 2.0f)) * (dY_46_v * dY_46_v)))))) * -0.5f;
	float tmp;
	if (t_6 >= t_3) {
		tmp = powf(fmaxf(t_6, t_3), -0.5f) * (floorf(w) * dX_46_u);
	} else {
		tmp = (coshf(t_8) + sinhf(t_8)) * (floorf(w) * 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 = floor(w) ^ Float32(2.0)
	t_1 = Float32(dY_46_u * Float32(t_0 * dY_46_u))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = Float32(Float32(Float32(dY_46_u * dY_46_u) * t_0) - Float32(Float32(Float32(Float32(-1.0) * dY_46_v) * dY_46_v) * t_2))
	t_4 = Float32(t_2 * dY_46_v)
	t_5 = Float32(t_2 * dX_46_v)
	t_6 = fma(t_5, dX_46_v, Float32(Float32(dX_46_u * dX_46_u) * t_0))
	t_7 = Float32(Float32(-1.0) * Float32(t_1 * Float32(-1.0)))
	t_8 = Float32(log(fmax(Float32(Float32(t_5 * dX_46_v) - Float32(Float32(Float32(t_0 * dX_46_u) * dX_46_u) * Float32(-1.0))), Float32(Float32((t_1 ^ Float32(3.0)) - Float32((Float32(Float32(-1.0) * dY_46_v) ^ Float32(3.0)) * (t_4 ^ Float32(3.0)))) / fma(t_7, t_7, Float32(fma(Float32(Float32(dY_46_u * dY_46_u) * Float32(-1.0)), Float32(t_0 * t_2), (t_4 ^ Float32(2.0))) * Float32(dY_46_v * dY_46_v)))))) * Float32(-0.5))
	tmp = Float32(0.0)
	if (t_6 >= t_3)
		tmp = Float32((fmax(t_6, t_3) ^ Float32(-0.5)) * Float32(floor(w) * dX_46_u));
	else
		tmp = Float32(Float32(cosh(t_8) + sinh(t_8)) * Float32(floor(w) * dY_46_u));
	end
	return tmp
end

Derivation

1. Initial program 78.0%

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

4. Applied rewrites 51.3%

   \[\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):\\ \;\;\;\;\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
5. Step-by-step derivation

   1. lift-*.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(\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), \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   2. pow2 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   3. lift-*.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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   4. 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(\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)}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   5. unpow-prod-down 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(\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 \right)}^{2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   6. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot {dX.v}^{2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   7. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2} \cdot e^{\log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   8. prod-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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   9. 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(\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) + e^{\log \left(\left\lfloor h\right\rfloor \right) \cdot 2 + \log dX.v \cdot 2}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   10. lower-fma.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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   11. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   12. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   13. 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(\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

   14. lower-log.f32 48.7

       \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

6. Applied rewrites 48.7%

   \[\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):\\ \;\;\;\;\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) + e^{\mathsf{fma}\left(\log \left(\left\lfloor h\right\rfloor \right), 2, \log dX.v \cdot 2\right)}, \frac{{\left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {dY.u}^{3} - {\left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)}^{3}}{\mathsf{fma}\left(\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.u, \mathsf{fma}\left(\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right) \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right), dY.v, \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right) \cdot \left(\left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(-dY.v\right)\right)\right)\right)}\right)}} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

7. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \sqrt{\frac{1}{\mathsf{max}\left(e^{2 \cdot \log dX.v + 2 \cdot \log \left(\left\lfloor h\right\rfloor \right)} + {dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \frac{{dY.u}^{6} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{6} - -1 \cdot \left({dY.v}^{6} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{6}\right)}{-1 \cdot \left({dY.u}^{2} \cdot \left({dY.v}^{2} \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right) + \left({dY.u}^{4} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{4} + {dY.v}^{4} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{4}\right)}\right)}}\\ } \end{array}} \]

9. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \frac{{\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{3} \cdot {\left(\left(dY.u \cdot dY.u\right) \cdot dY.u\right)}^{2} - {\left(\left(-1 \cdot dY.v\right) \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(dY.u \cdot dY.u\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left({\left({\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}^{2}, {\left(dY.v \cdot dY.v\right)}^{2}, {\left({\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}^{2} \cdot {\left(dY.u \cdot dY.u\right)}^{2}\right)\right)}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ } \end{array}} \]

11. Applied rewrites 47.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh \left(\log \left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{3}}{\mathsf{fma}\left(-\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1, -\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1, \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right) \cdot -0.5\right) + \sinh \left(\log \left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{3}}{\mathsf{fma}\left(-\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1, -\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1, \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right) \cdot -0.5\right)\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]

12. Final simplification 47.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \geq \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}:\\ \;\;\;\;{\left(\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, dX.v, \left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \left(dY.u \cdot dY.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} - \left(\left(-1 \cdot dY.v\right) \cdot dY.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{-0.5} \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh \left(\log \left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1\right), -1 \cdot \left(\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1\right), \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right) \cdot -0.5\right) + \sinh \left(\log \left(\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v - \left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u\right) \cdot -1, \frac{{\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right)}^{3} - {\left(-1 \cdot dY.v\right)}^{3} \cdot {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{3}}{\mathsf{fma}\left(-1 \cdot \left(\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1\right), -1 \cdot \left(\left(dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)\right) \cdot -1\right), \mathsf{fma}\left(\left(dY.u \cdot dY.u\right) \cdot -1, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, {\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}^{2}\right) \cdot \left(dY.v \cdot dY.v\right)\right)}\right)\right) \cdot -0.5\right)\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)\\ \end{array} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))