Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 75.3% → 75.4%

Time: 16.2s

Alternatives: 5

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_1 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_3 := {t\_2}^{2} + {t\_1}^{2}\\ t_4 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_5 := {t\_4}^{2}\\ \mathbf{if}\;t\_4 \cdot t\_4 + t\_0 \cdot t\_0 \geq t\_1 \cdot t\_1 + t\_2 \cdot t\_2:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({t\_0}^{2} + t\_5, t\_3\right)}}{t\_4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + t\_5, t\_3\right)\right)}^{0.5}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.u (floor w)))
        (t_1 (* dY.v (floor h)))
        (t_2 (* dY.u (floor w)))
        (t_3 (+ (pow t_2 2.0) (pow t_1 2.0)))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow t_4 2.0)))
   (if (>= (+ (* t_4 t_4) (* t_0 t_0)) (+ (* t_1 t_1) (* t_2 t_2)))
     (/ 1.0 (/ (sqrt (fmax (+ (pow t_0 2.0) t_5) t_3)) t_4))
     (* (/ 1.0 (pow (fmax (+ (exp (/ 0.0 0.0)) t_5) t_3) 0.5)) 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 = dX_46_u * floorf(w);
	float t_1 = dY_46_v * floorf(h);
	float t_2 = dY_46_u * floorf(w);
	float t_3 = powf(t_2, 2.0f) + powf(t_1, 2.0f);
	float t_4 = dX_46_v * floorf(h);
	float t_5 = powf(t_4, 2.0f);
	float tmp;
	if (((t_4 * t_4) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2))) {
		tmp = 1.0f / (sqrtf(fmaxf((powf(t_0, 2.0f) + t_5), t_3)) / t_4);
	} else {
		tmp = (1.0f / powf(fmaxf((expf((0.0f / 0.0f)) + t_5), t_3), 0.5f)) * 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(dX_46_u * floor(w))
	t_1 = Float32(dY_46_v * floor(h))
	t_2 = Float32(dY_46_u * floor(w))
	t_3 = Float32((t_2 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = t_4 ^ Float32(2.0)
	tmp = Float32(0.0)
	if (Float32(Float32(t_4 * t_4) + Float32(t_0 * t_0)) >= Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)))
		tmp = Float32(Float32(1.0) / Float32(sqrt(((Float32((t_0 ^ Float32(2.0)) + t_5) != Float32((t_0 ^ Float32(2.0)) + t_5)) ? t_3 : ((t_3 != t_3) ? Float32((t_0 ^ Float32(2.0)) + t_5) : max(Float32((t_0 ^ Float32(2.0)) + t_5), t_3)))) / t_4));
	else
		tmp = Float32(Float32(Float32(1.0) / (((Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5) != Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5)) ? t_3 : ((t_3 != t_3) ? Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5) : max(Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5), t_3))) ^ Float32(0.5))) * 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 = dX_46_u * floor(w);
	t_1 = dY_46_v * floor(h);
	t_2 = dY_46_u * floor(w);
	t_3 = (t_2 ^ single(2.0)) + (t_1 ^ single(2.0));
	t_4 = dX_46_v * floor(h);
	t_5 = t_4 ^ single(2.0);
	tmp = single(0.0);
	if (((t_4 * t_4) + (t_0 * t_0)) >= ((t_1 * t_1) + (t_2 * t_2)))
		tmp = single(1.0) / (sqrt(max(((t_0 ^ single(2.0)) + t_5), t_3)) / t_4);
	else
		tmp = (single(1.0) / (max((exp((single(0.0) / single(0.0))) + t_5), t_3) ^ single(0.5))) * t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

4. Applied rewrites 78.4%

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

6. Applied rewrites 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

7. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \geq \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array} \]
8. Add Preprocessing

Alternative 2: 68.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := t\_0 \cdot t\_0 + t\_1 \cdot t\_1\\ t_3 := {t\_1}^{2} + {t\_0}^{2}\\ t_4 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_5 := {t\_4}^{2}\\ t_6 := \frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + t\_5, t\_3\right)\right)}^{0.5}} \cdot t\_0\\ t_7 := \frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + t\_5, t\_3\right)}}{t\_4}}\\ \mathbf{if}\;dX.u \leq 12:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_5 \geq t\_2:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array}\\ \mathbf{elif}\;\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u \geq t\_2:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h)))
        (t_1 (* dY.u (floor w)))
        (t_2 (+ (* t_0 t_0) (* t_1 t_1)))
        (t_3 (+ (pow t_1 2.0) (pow t_0 2.0)))
        (t_4 (* dX.v (floor h)))
        (t_5 (pow t_4 2.0))
        (t_6 (* (/ 1.0 (pow (fmax (+ (exp (/ 0.0 0.0)) t_5) t_3) 0.5)) t_0))
        (t_7
         (/
          1.0
          (/ (sqrt (fmax (+ (pow (* dX.u (floor w)) 2.0) t_5) t_3)) t_4))))
   (if (<= dX.u 12.0)
     (if (>= t_5 t_2) t_7 t_6)
     (if (>= (* (* (pow (floor w) 2.0) dX.u) dX.u) t_2) t_7 t_6))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = dY_46_u * floorf(w);
	float t_2 = (t_0 * t_0) + (t_1 * t_1);
	float t_3 = powf(t_1, 2.0f) + powf(t_0, 2.0f);
	float t_4 = dX_46_v * floorf(h);
	float t_5 = powf(t_4, 2.0f);
	float t_6 = (1.0f / powf(fmaxf((expf((0.0f / 0.0f)) + t_5), t_3), 0.5f)) * t_0;
	float t_7 = 1.0f / (sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + t_5), t_3)) / t_4);
	float tmp_1;
	if (dX_46_u <= 12.0f) {
		float tmp_2;
		if (t_5 >= t_2) {
			tmp_2 = t_7;
		} else {
			tmp_2 = t_6;
		}
		tmp_1 = tmp_2;
	} else if (((powf(floorf(w), 2.0f) * dX_46_u) * dX_46_u) >= t_2) {
		tmp_1 = t_7;
	} else {
		tmp_1 = t_6;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))
	t_3 = Float32((t_1 ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))
	t_4 = Float32(dX_46_v * floor(h))
	t_5 = t_4 ^ Float32(2.0)
	t_6 = Float32(Float32(Float32(1.0) / (((Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5) != Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5)) ? t_3 : ((t_3 != t_3) ? Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5) : max(Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_5), t_3))) ^ Float32(0.5))) * t_0)
	t_7 = Float32(Float32(1.0) / Float32(sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_5) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_5)) ? t_3 : ((t_3 != t_3) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_5) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_5), t_3)))) / t_4))
	tmp_1 = Float32(0.0)
	if (dX_46_u <= Float32(12.0))
		tmp_2 = Float32(0.0)
		if (t_5 >= t_2)
			tmp_2 = t_7;
		else
			tmp_2 = t_6;
		end
		tmp_1 = tmp_2;
	elseif (Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u) * dX_46_u) >= t_2)
		tmp_1 = t_7;
	else
		tmp_1 = t_6;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = dY_46_v * floor(h);
	t_1 = dY_46_u * floor(w);
	t_2 = (t_0 * t_0) + (t_1 * t_1);
	t_3 = (t_1 ^ single(2.0)) + (t_0 ^ single(2.0));
	t_4 = dX_46_v * floor(h);
	t_5 = t_4 ^ single(2.0);
	t_6 = (single(1.0) / (max((exp((single(0.0) / single(0.0))) + t_5), t_3) ^ single(0.5))) * t_0;
	t_7 = single(1.0) / (sqrt(max((((dX_46_u * floor(w)) ^ single(2.0)) + t_5), t_3)) / t_4);
	tmp_2 = single(0.0);
	if (dX_46_u <= single(12.0))
		tmp_3 = single(0.0);
		if (t_5 >= t_2)
			tmp_3 = t_7;
		else
			tmp_3 = t_6;
		end
		tmp_2 = tmp_3;
	elseif ((((floor(w) ^ single(2.0)) * dX_46_u) * dX_46_u) >= t_2)
		tmp_2 = t_7;
	else
		tmp_2 = t_6;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 12

   1. Initial program 79.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

   4. Applied rewrites 79.4%

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

   6. Applied rewrites 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.v \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      3. unswap-sqr N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      4. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      5. lower-pow.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      7. lower-floor.f32 72.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   9. Applied rewrites 72.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   if 12 < dX.u

   1. Initial program 75.1%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

   4. Applied rewrites 75.2%

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

   6. Applied rewrites 75.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. Taylor expanded in dX.u around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      3. associate-*r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      4. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      5. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)} \cdot dX.u \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)} \cdot dX.u \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      7. lower-pow.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right) \cdot dX.u \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      8. lower-floor.f32 73.6

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right) \cdot dX.u \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   9. Applied rewrites 73.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right) \cdot dX.u} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 12:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array}\\ \mathbf{elif}\;\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u \geq \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_1 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_2 := {t\_1}^{2}\\ t_3 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_4 := {t\_0}^{2} + {t\_3}^{2}\\ \mathbf{if}\;t\_2 \geq t\_3 \cdot t\_3 + t\_0 \cdot t\_0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + t\_2, t\_4\right)}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + t\_2, t\_4\right)\right)}^{0.5}} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.u (floor w)))
        (t_1 (* dX.v (floor h)))
        (t_2 (pow t_1 2.0))
        (t_3 (* dY.v (floor h)))
        (t_4 (+ (pow t_0 2.0) (pow t_3 2.0))))
   (if (>= t_2 (+ (* t_3 t_3) (* t_0 t_0)))
     (/ 1.0 (/ (sqrt (fmax (+ (pow (* dX.u (floor w)) 2.0) t_2) t_4)) t_1))
     (* (/ 1.0 (pow (fmax (+ (exp (/ 0.0 0.0)) t_2) t_4) 0.5)) 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 = dY_46_u * floorf(w);
	float t_1 = dX_46_v * floorf(h);
	float t_2 = powf(t_1, 2.0f);
	float t_3 = dY_46_v * floorf(h);
	float t_4 = powf(t_0, 2.0f) + powf(t_3, 2.0f);
	float tmp;
	if (t_2 >= ((t_3 * t_3) + (t_0 * t_0))) {
		tmp = 1.0f / (sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + t_2), t_4)) / t_1);
	} else {
		tmp = (1.0f / powf(fmaxf((expf((0.0f / 0.0f)) + t_2), t_4), 0.5f)) * 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(dY_46_u * floor(w))
	t_1 = Float32(dX_46_v * floor(h))
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32(dY_46_v * floor(h))
	t_4 = Float32((t_0 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (t_2 >= Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)))
		tmp = Float32(Float32(1.0) / Float32(sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2)) ? t_4 : ((t_4 != t_4) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2), t_4)))) / t_1));
	else
		tmp = Float32(Float32(Float32(1.0) / (((Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2) != Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2)) ? t_4 : ((t_4 != t_4) ? Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2) : max(Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2), t_4))) ^ Float32(0.5))) * 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 = dY_46_u * floor(w);
	t_1 = dX_46_v * floor(h);
	t_2 = t_1 ^ single(2.0);
	t_3 = dY_46_v * floor(h);
	t_4 = (t_0 ^ single(2.0)) + (t_3 ^ single(2.0));
	tmp = single(0.0);
	if (t_2 >= ((t_3 * t_3) + (t_0 * t_0)))
		tmp = single(1.0) / (sqrt(max((((dX_46_u * floor(w)) ^ single(2.0)) + t_2), t_4)) / t_1);
	else
		tmp = (single(1.0) / (max((exp((single(0.0) / single(0.0))) + t_2), t_4) ^ single(0.5))) * t_3;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

4. Applied rewrites 78.4%

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

6. Applied rewrites 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

7. Taylor expanded in dX.u around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   2. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.v \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   3. unswap-sqr N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   4. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   5. lower-pow.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;{\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. lower-floor.f32 69.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

9. Applied rewrites 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

10. Final simplification 69.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array} \]
11. Add Preprocessing

Alternative 4: 60.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {t\_0}^{2}\\ t_2 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_3 := {t\_2}^{2}\\ t_4 := \frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + t\_3, t\_1\right)}}{t\_2}}\\ t_5 := \frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + t\_3, t\_1\right)\right)}^{0.5}} \cdot t\_0\\ \mathbf{if}\;dY.u \leq 1400000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array}\\ \mathbf{elif}\;t\_3 \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h)))
        (t_1 (+ (pow (* dY.u (floor w)) 2.0) (pow t_0 2.0)))
        (t_2 (* dX.v (floor h)))
        (t_3 (pow t_2 2.0))
        (t_4
         (/
          1.0
          (/ (sqrt (fmax (+ (pow (* dX.u (floor w)) 2.0) t_3) t_1)) t_2)))
        (t_5 (* (/ 1.0 (pow (fmax (+ (exp (/ 0.0 0.0)) t_3) t_1) 0.5)) t_0)))
   (if (<= dY.u 1400000.0)
     (if (>= t_3 (* (* (pow (floor h) 2.0) dY.v) dY.v)) t_4 t_5)
     (if (>= t_3 (* (* (pow (floor w) 2.0) dY.u) dY.u)) t_4 t_5))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = powf((dY_46_u * floorf(w)), 2.0f) + powf(t_0, 2.0f);
	float t_2 = dX_46_v * floorf(h);
	float t_3 = powf(t_2, 2.0f);
	float t_4 = 1.0f / (sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + t_3), t_1)) / t_2);
	float t_5 = (1.0f / powf(fmaxf((expf((0.0f / 0.0f)) + t_3), t_1), 0.5f)) * t_0;
	float tmp_1;
	if (dY_46_u <= 1400000.0f) {
		float tmp_2;
		if (t_3 >= ((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v)) {
			tmp_2 = t_4;
		} else {
			tmp_2 = t_5;
		}
		tmp_1 = tmp_2;
	} else if (t_3 >= ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u)) {
		tmp_1 = t_4;
	} else {
		tmp_1 = t_5;
	}
	return tmp_1;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = t_2 ^ Float32(2.0)
	t_4 = Float32(Float32(1.0) / Float32(sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_3) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_3)) ? t_1 : ((t_1 != t_1) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_3) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_3), t_1)))) / t_2))
	t_5 = Float32(Float32(Float32(1.0) / (((Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_3) != Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_3)) ? t_1 : ((t_1 != t_1) ? Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_3) : max(Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_3), t_1))) ^ Float32(0.5))) * t_0)
	tmp_1 = Float32(0.0)
	if (dY_46_u <= Float32(1400000.0))
		tmp_2 = Float32(0.0)
		if (t_3 >= Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v))
			tmp_2 = t_4;
		else
			tmp_2 = t_5;
		end
		tmp_1 = tmp_2;
	elseif (t_3 >= Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u))
		tmp_1 = t_4;
	else
		tmp_1 = t_5;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = dY_46_v * floor(h);
	t_1 = ((dY_46_u * floor(w)) ^ single(2.0)) + (t_0 ^ single(2.0));
	t_2 = dX_46_v * floor(h);
	t_3 = t_2 ^ single(2.0);
	t_4 = single(1.0) / (sqrt(max((((dX_46_u * floor(w)) ^ single(2.0)) + t_3), t_1)) / t_2);
	t_5 = (single(1.0) / (max((exp((single(0.0) / single(0.0))) + t_3), t_1) ^ single(0.5))) * t_0;
	tmp_2 = single(0.0);
	if (dY_46_u <= single(1400000.0))
		tmp_3 = single(0.0);
		if (t_3 >= (((floor(h) ^ single(2.0)) * dY_46_v) * dY_46_v))
			tmp_3 = t_4;
		else
			tmp_3 = t_5;
		end
		tmp_2 = tmp_3;
	elseif (t_3 >= (((floor(w) ^ single(2.0)) * dY_46_u) * dY_46_u))
		tmp_2 = t_4;
	else
		tmp_2 = t_5;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1.4e6

   1. Initial program 79.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

   4. Applied rewrites 79.4%

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

   6. Applied rewrites 79.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.v \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      3. unswap-sqr N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      4. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      5. lower-pow.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      7. lower-floor.f32 70.4

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   9. Applied rewrites 70.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   10. Taylor expanded in dY.v around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       2. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       3. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       4. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       6. lower-pow.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       7. lower-floor.f32 66.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   12. Applied rewrites 66.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   if 1.4e6 < dY.u

   1. Initial program 73.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

   4. Applied rewrites 73.1%

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

   6. Applied rewrites 73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. Taylor expanded in dX.u around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      2. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.v \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      3. unswap-sqr N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      4. unpow2 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      5. lower-pow.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      6. lower-*.f32 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

      7. lower-floor.f32 62.3

         \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   9. Applied rewrites 62.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   10. Taylor expanded in dY.v around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       2. unpow2 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       3. associate-*r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       4. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       5. lower-*.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       6. lower-pow.f32 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

       7. lower-floor.f32 62.3

          \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   12. Applied rewrites 62.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 1400000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array}\\ \mathbf{elif}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_2 := {t\_1}^{2}\\ t_3 := {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {t\_0}^{2}\\ \mathbf{if}\;t\_2 \geq \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + t\_2, t\_3\right)}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + t\_2, t\_3\right)\right)}^{0.5}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h)))
        (t_1 (* dX.v (floor h)))
        (t_2 (pow t_1 2.0))
        (t_3 (+ (pow (* dY.u (floor w)) 2.0) (pow t_0 2.0))))
   (if (>= t_2 (* (* (pow (floor h) 2.0) dY.v) dY.v))
     (/ 1.0 (/ (sqrt (fmax (+ (pow (* dX.u (floor w)) 2.0) t_2) t_3)) t_1))
     (* (/ 1.0 (pow (fmax (+ (exp (/ 0.0 0.0)) t_2) t_3) 0.5)) t_0))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = dX_46_v * floorf(h);
	float t_2 = powf(t_1, 2.0f);
	float t_3 = powf((dY_46_u * floorf(w)), 2.0f) + powf(t_0, 2.0f);
	float tmp;
	if (t_2 >= ((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v)) {
		tmp = 1.0f / (sqrtf(fmaxf((powf((dX_46_u * floorf(w)), 2.0f) + t_2), t_3)) / t_1);
	} else {
		tmp = (1.0f / powf(fmaxf((expf((0.0f / 0.0f)) + t_2), t_3), 0.5f)) * t_0;
	}
	return tmp;
}

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dX_46_v * floor(h))
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (t_2 >= Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v))
		tmp = Float32(Float32(1.0) / Float32(sqrt(((Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2) != Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2)) ? t_3 : ((t_3 != t_3) ? Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2) : max(Float32((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) + t_2), t_3)))) / t_1));
	else
		tmp = Float32(Float32(Float32(1.0) / (((Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2) != Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2)) ? t_3 : ((t_3 != t_3) ? Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2) : max(Float32(exp(Float32(Float32(0.0) / Float32(0.0))) + t_2), t_3))) ^ Float32(0.5))) * t_0);
	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 = dY_46_v * floor(h);
	t_1 = dX_46_v * floor(h);
	t_2 = t_1 ^ single(2.0);
	t_3 = ((dY_46_u * floor(w)) ^ single(2.0)) + (t_0 ^ single(2.0));
	tmp = single(0.0);
	if (t_2 >= (((floor(h) ^ single(2.0)) * dY_46_v) * dY_46_v))
		tmp = single(1.0) / (sqrt(max((((dX_46_u * floor(w)) ^ single(2.0)) + t_2), t_3)) / t_1);
	else
		tmp = (single(1.0) / (max((exp((single(0.0) / single(0.0))) + t_2), t_3) ^ single(0.5))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 78.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

4. Applied rewrites 78.4%

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

6. Applied rewrites 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

7. Taylor expanded in dX.u around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   2. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\left(dX.v \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   3. unswap-sqr N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right)} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   4. unpow2 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   5. lower-pow.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   6. lower-*.f32 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;{\color{blue}{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

   7. lower-floor.f32 69.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \color{blue}{\left\lfloor h\right\rfloor }\right)}^{2} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

9. Applied rewrites 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\color{blue}{{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}} \geq \left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right):\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

10. Taylor expanded in dY.v around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    2. unpow2 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    3. associate-*r* N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    4. lower-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    5. lower-*.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    6. lower-pow.f32 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{\frac{1}{2}}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

    7. lower-floor.f32 63.7

       \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

12. Applied rewrites 63.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\\ \end{array} \]

13. Final simplification 63.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}}{dX.v \cdot \left\lfloor h\right\rfloor }}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{max}\left(e^{\frac{0}{0}} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}, {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}^{0.5}} \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right)\\ \end{array} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, v)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dX.v)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dY.v))))