Anisotropic x16 LOD (line direction, u)

Percentage Accurate: 76.4% → 76.6%

Time: 41.7s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((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_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, h, dX_46_u, dX_46_v, dY_46_u, dY_46_v, maxAniso)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * dX_46_u)
	t_3 = Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0))
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(Float32(t_1 * t_1) + Float32(t_4 * t_4))
	t_6 = Float32(Float32(1.0) / sqrt(((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_2);
	else
		tmp = Float32(t_6 * t_1);
	end
	return tmp
end

MATLAB

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

Alternative 1: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.5%

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

   1. add-log-exp 59.0%

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

4. Applied egg-rr 59.0%

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

6. Applied egg-rr 79.6%

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

7. Taylor expanded in w around 0 79.6%

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

9. Simplified 79.6%

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

    1. pow2 79.6%

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

11. Applied egg-rr 79.6%

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

12. Final simplification 79.6%

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

Alternative 2: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := {\left(\mathsf{hypot}\left(t\_1, t\_0\right)\right)}^{2}\\ t_3 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_4 := {\left(\mathsf{hypot}\left(t\_3, dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}\\ t_5 := \left\lfloor w\right\rfloor \cdot \left(dY.u \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_4, t\_2\right)}}\right)\\ t_6 := {t\_0}^{2}\\ \mathbf{if}\;dY.v \leq 50:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_4 \geq {t\_1}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2}, {dX.v}^{2}, {t\_3}^{2}\right), t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array}\\ \mathbf{elif}\;t\_4 \geq t\_6:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_4, t\_6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor w) dY.u))
        (t_2 (pow (hypot t_1 t_0) 2.0))
        (t_3 (* dX.u (floor w)))
        (t_4 (pow (hypot t_3 (* dX.v (floor h))) 2.0))
        (t_5 (* (floor w) (* dY.u (sqrt (/ 1.0 (fmax t_4 t_2))))))
        (t_6 (pow t_0 2.0)))
   (if (<= dY.v 50.0)
     (if (>= t_4 (pow t_1 2.0))
       (*
        dX.u
        (*
         (floor w)
         (sqrt
          (/
           1.0
           (fmax
            (fma (pow (floor h) 2.0) (pow dX.v 2.0) (pow t_3 2.0))
            t_2)))))
       t_5)
     (if (>= t_4 t_6)
       (* dX.u (* (floor w) (sqrt (/ 1.0 (fmax t_4 t_6)))))
       t_5))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = powf(hypotf(t_1, t_0), 2.0f);
	float t_3 = dX_46_u * floorf(w);
	float t_4 = powf(hypotf(t_3, (dX_46_v * floorf(h))), 2.0f);
	float t_5 = floorf(w) * (dY_46_u * sqrtf((1.0f / fmaxf(t_4, t_2))));
	float t_6 = powf(t_0, 2.0f);
	float tmp_1;
	if (dY_46_v <= 50.0f) {
		float tmp_2;
		if (t_4 >= powf(t_1, 2.0f)) {
			tmp_2 = dX_46_u * (floorf(w) * sqrtf((1.0f / fmaxf(fmaf(powf(floorf(h), 2.0f), powf(dX_46_v, 2.0f), powf(t_3, 2.0f)), t_2))));
		} else {
			tmp_2 = t_5;
		}
		tmp_1 = tmp_2;
	} else if (t_4 >= t_6) {
		tmp_1 = dX_46_u * (floorf(w) * sqrtf((1.0f / fmaxf(t_4, t_6))));
	} 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(floor(h) * dY_46_v)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = hypot(t_1, t_0) ^ Float32(2.0)
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = hypot(t_3, Float32(dX_46_v * floor(h))) ^ Float32(2.0)
	t_5 = Float32(floor(w) * Float32(dY_46_u * sqrt(Float32(Float32(1.0) / ((t_4 != t_4) ? t_2 : ((t_2 != t_2) ? t_4 : max(t_4, t_2)))))))
	t_6 = t_0 ^ Float32(2.0)
	tmp_1 = Float32(0.0)
	if (dY_46_v <= Float32(50.0))
		tmp_2 = Float32(0.0)
		if (t_4 >= (t_1 ^ Float32(2.0)))
			tmp_2 = Float32(dX_46_u * Float32(floor(w) * sqrt(Float32(Float32(1.0) / ((fma((floor(h) ^ Float32(2.0)), (dX_46_v ^ Float32(2.0)), (t_3 ^ Float32(2.0))) != fma((floor(h) ^ Float32(2.0)), (dX_46_v ^ Float32(2.0)), (t_3 ^ Float32(2.0)))) ? t_2 : ((t_2 != t_2) ? fma((floor(h) ^ Float32(2.0)), (dX_46_v ^ Float32(2.0)), (t_3 ^ Float32(2.0))) : max(fma((floor(h) ^ Float32(2.0)), (dX_46_v ^ Float32(2.0)), (t_3 ^ Float32(2.0))), t_2)))))));
		else
			tmp_2 = t_5;
		end
		tmp_1 = tmp_2;
	elseif (t_4 >= t_6)
		tmp_1 = Float32(dX_46_u * Float32(floor(w) * sqrt(Float32(Float32(1.0) / ((t_4 != t_4) ? t_6 : ((t_6 != t_6) ? t_4 : max(t_4, t_6)))))));
	else
		tmp_1 = t_5;
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 50

   1. Initial program 79.3%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in w around 0 79.2%

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

   7. Simplified 78.9%

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

      1. *-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. unpow2 78.9%

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

      4. hypot-undefine 78.9%

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

      5. hypot-undefine 78.9%

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

      6. add-sqr-sqrt 79.0%

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

      7. +-commutative 79.0%

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

      8. swap-sqr 79.1%

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

      9. fma-define 79.1%

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

      10. pow2 79.1%

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

      11. pow2 79.1%

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

      12. pow2 79.1%

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

   9. Applied egg-rr 79.1%

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

   10. Taylor expanded in dY.u around inf 73.5%

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

       1. *-commutative 73.5%

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

       2. unpow2 73.5%

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

       3. unpow2 73.5%

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

       4. swap-sqr 73.5%

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

       5. unpow2 73.5%

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

   12. Simplified 73.5%

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

   if 50 < dY.v

   1. Initial program 80.0%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in w around 0 80.0%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in dY.u around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow2 75.9%

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

      3. unpow2 75.9%

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

      4. swap-sqr 75.9%

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

      5. unpow2 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in dY.u around 0 75.9%

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

       1. *-commutative 75.9%

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

       2. unpow2 75.9%

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

       3. unpow2 75.9%

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

       4. swap-sqr 75.9%

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

       5. unpow2 75.9%

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

   13. Simplified 75.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 50:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2} \geq {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left(\mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2}, {dX.v}^{2}, {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot \sqrt{\frac{1}{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \end{array}\\ \mathbf{elif}\;{\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2} \geq {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot \sqrt{\frac{1}{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.3%

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

6. Applied egg-rr 79.5%

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

7. Taylor expanded in w around 0 79.3%

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

9. Simplified 79.5%

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

Alternative 4: 69.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := {t\_0}^{2}\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_3 := {\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}\\ t_4 := \sqrt{\frac{1}{\mathsf{max}\left(t\_3, {\left(\mathsf{hypot}\left(t\_2, t\_0\right)\right)}^{2}\right)}}\\ t_5 := \left\lfloor w\right\rfloor \cdot \left(dY.u \cdot t\_4\right)\\ \mathbf{if}\;dY.v \leq 5.099999904632568:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t\_3 \geq {t\_2}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array}\\ \mathbf{elif}\;t\_3 \geq t\_1:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left(t\_3, t\_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (pow t_0 2.0))
        (t_2 (* (floor w) dY.u))
        (t_3 (pow (hypot (* dX.u (floor w)) (* dX.v (floor h))) 2.0))
        (t_4 (sqrt (/ 1.0 (fmax t_3 (pow (hypot t_2 t_0) 2.0)))))
        (t_5 (* (floor w) (* dY.u t_4))))
   (if (<= dY.v 5.099999904632568)
     (if (>= t_3 (pow t_2 2.0)) (* dX.u (* (floor w) t_4)) t_5)
     (if (>= t_3 t_1)
       (* dX.u (* (floor w) (sqrt (/ 1.0 (fmax t_3 t_1)))))
       t_5))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = powf(t_0, 2.0f);
	float t_2 = floorf(w) * dY_46_u;
	float t_3 = powf(hypotf((dX_46_u * floorf(w)), (dX_46_v * floorf(h))), 2.0f);
	float t_4 = sqrtf((1.0f / fmaxf(t_3, powf(hypotf(t_2, t_0), 2.0f))));
	float t_5 = floorf(w) * (dY_46_u * t_4);
	float tmp_1;
	if (dY_46_v <= 5.099999904632568f) {
		float tmp_2;
		if (t_3 >= powf(t_2, 2.0f)) {
			tmp_2 = dX_46_u * (floorf(w) * t_4);
		} else {
			tmp_2 = t_5;
		}
		tmp_1 = tmp_2;
	} else if (t_3 >= t_1) {
		tmp_1 = dX_46_u * (floorf(w) * sqrtf((1.0f / fmaxf(t_3, t_1))));
	} 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(floor(h) * dY_46_v)
	t_1 = t_0 ^ Float32(2.0)
	t_2 = Float32(floor(w) * dY_46_u)
	t_3 = hypot(Float32(dX_46_u * floor(w)), Float32(dX_46_v * floor(h))) ^ Float32(2.0)
	t_4 = sqrt(Float32(Float32(1.0) / ((t_3 != t_3) ? (hypot(t_2, t_0) ^ Float32(2.0)) : (((hypot(t_2, t_0) ^ Float32(2.0)) != (hypot(t_2, t_0) ^ Float32(2.0))) ? t_3 : max(t_3, (hypot(t_2, t_0) ^ Float32(2.0)))))))
	t_5 = Float32(floor(w) * Float32(dY_46_u * t_4))
	tmp_1 = Float32(0.0)
	if (dY_46_v <= Float32(5.099999904632568))
		tmp_2 = Float32(0.0)
		if (t_3 >= (t_2 ^ Float32(2.0)))
			tmp_2 = Float32(dX_46_u * Float32(floor(w) * t_4));
		else
			tmp_2 = t_5;
		end
		tmp_1 = tmp_2;
	elseif (t_3 >= t_1)
		tmp_1 = Float32(dX_46_u * Float32(floor(w) * sqrt(Float32(Float32(1.0) / ((t_3 != t_3) ? t_1 : ((t_1 != t_1) ? t_3 : max(t_3, t_1)))))));
	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 = floor(h) * dY_46_v;
	t_1 = t_0 ^ single(2.0);
	t_2 = floor(w) * dY_46_u;
	t_3 = hypot((dX_46_u * floor(w)), (dX_46_v * floor(h))) ^ single(2.0);
	t_4 = sqrt((single(1.0) / max(t_3, (hypot(t_2, t_0) ^ single(2.0)))));
	t_5 = floor(w) * (dY_46_u * t_4);
	tmp_2 = single(0.0);
	if (dY_46_v <= single(5.099999904632568))
		tmp_3 = single(0.0);
		if (t_3 >= (t_2 ^ single(2.0)))
			tmp_3 = dX_46_u * (floor(w) * t_4);
		else
			tmp_3 = t_5;
		end
		tmp_2 = tmp_3;
	elseif (t_3 >= t_1)
		tmp_2 = dX_46_u * (floor(w) * sqrt((single(1.0) / max(t_3, t_1))));
	else
		tmp_2 = t_5;
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 5.0999999

   1. Initial program 79.1%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in w around 0 79.0%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in dY.u around inf 73.1%

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

      1. *-commutative 73.2%

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

      2. unpow2 73.2%

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

      3. unpow2 73.2%

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

      4. swap-sqr 73.2%

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

      5. unpow2 73.2%

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

   10. Simplified 73.1%

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

   if 5.0999999 < dY.v

   1. Initial program 80.8%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in w around 0 80.7%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in dY.u around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. unpow2 76.8%

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

      4. swap-sqr 76.8%

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

      5. unpow2 76.8%

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

   10. Simplified 76.8%

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

   11. Taylor expanded in dY.u around 0 76.8%

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

       1. *-commutative 76.8%

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

       2. unpow2 76.8%

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

       3. unpow2 76.8%

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

       4. swap-sqr 76.8%

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

       5. unpow2 76.8%

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

   13. Simplified 76.8%

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

Alternative 5: 68.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.6%

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

5. Taylor expanded in w around 0 79.3%

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

7. Simplified 79.1%

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

8. Taylor expanded in dY.u around 0 69.5%

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

   1. *-commutative 69.5%

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

   2. unpow2 69.5%

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

   3. unpow2 69.5%

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

   4. swap-sqr 69.5%

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

   5. unpow2 69.5%

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

10. Simplified 69.5%

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

11. Taylor expanded in dY.u around 0 72.2%

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

    1. *-commutative 69.5%

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

    2. unpow2 69.5%

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

    3. unpow2 69.5%

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

    4. swap-sqr 69.5%

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

    5. unpow2 69.5%

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

13. Simplified 72.2%

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

Alternative 6: 65.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.6%

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

5. Taylor expanded in w around 0 79.3%

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

7. Simplified 79.1%

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

8. Taylor expanded in dY.u around 0 69.5%

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

   1. *-commutative 69.5%

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

   2. unpow2 69.5%

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

   3. unpow2 69.5%

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

   4. swap-sqr 69.5%

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

   5. unpow2 69.5%

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

10. Simplified 69.5%

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

11. Taylor expanded in dX.u around 0 69.7%

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

13. Simplified 70.0%

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

Alternative 7: 61.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_2 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_3 := \mathsf{max}\left({\left(\mathsf{hypot}\left(t\_1, t\_2\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_0\right)\right)}^{2}\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \sqrt{\frac{1}{t\_3}}\\ \mathbf{if}\;dX.v \leq 0.0010000000474974513:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_1}^{2} \geq {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}:\\ \;\;\;\;dX.u \cdot \frac{\left\lfloor w\right\rfloor }{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \frac{dY.u}{t\_4}\\ \end{array}\\ \mathbf{elif}\;{t\_2}^{2} \geq {t\_0}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot t\_5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* dX.u (floor w)))
        (t_2 (* dX.v (floor h)))
        (t_3
         (fmax
          (pow (hypot t_1 t_2) 2.0)
          (pow (hypot (* (floor w) dY.u) t_0) 2.0)))
        (t_4 (sqrt t_3))
        (t_5 (sqrt (/ 1.0 t_3))))
   (if (<= dX.v 0.0010000000474974513)
     (if (>= (pow t_1 2.0) (* (pow (floor h) 2.0) (pow dY.v 2.0)))
       (* dX.u (/ (floor w) t_4))
       (* (floor w) (/ dY.u t_4)))
     (if (>= (pow t_2 2.0) (pow t_0 2.0))
       (* dX.u (* (floor w) t_5))
       (* (floor w) (* dY.u t_5))))))

C

float code(float w, float h, float dX_46_u, float dX_46_v, float dY_46_u, float dY_46_v, float maxAniso) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = dX_46_u * floorf(w);
	float t_2 = dX_46_v * floorf(h);
	float t_3 = fmaxf(powf(hypotf(t_1, t_2), 2.0f), powf(hypotf((floorf(w) * dY_46_u), t_0), 2.0f));
	float t_4 = sqrtf(t_3);
	float t_5 = sqrtf((1.0f / t_3));
	float tmp_1;
	if (dX_46_v <= 0.0010000000474974513f) {
		float tmp_2;
		if (powf(t_1, 2.0f) >= (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f))) {
			tmp_2 = dX_46_u * (floorf(w) / t_4);
		} else {
			tmp_2 = floorf(w) * (dY_46_u / t_4);
		}
		tmp_1 = tmp_2;
	} else if (powf(t_2, 2.0f) >= powf(t_0, 2.0f)) {
		tmp_1 = dX_46_u * (floorf(w) * t_5);
	} else {
		tmp_1 = floorf(w) * (dY_46_u * 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(floor(h) * dY_46_v)
	t_1 = Float32(dX_46_u * floor(w))
	t_2 = Float32(dX_46_v * floor(h))
	t_3 = ((hypot(t_1, t_2) ^ Float32(2.0)) != (hypot(t_1, t_2) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0))) ? (hypot(t_1, t_2) ^ Float32(2.0)) : max((hypot(t_1, t_2) ^ Float32(2.0)), (hypot(Float32(floor(w) * dY_46_u), t_0) ^ Float32(2.0))))
	t_4 = sqrt(t_3)
	t_5 = sqrt(Float32(Float32(1.0) / t_3))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(0.0010000000474974513))
		tmp_2 = Float32(0.0)
		if ((t_1 ^ Float32(2.0)) >= Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))
			tmp_2 = Float32(dX_46_u * Float32(floor(w) / t_4));
		else
			tmp_2 = Float32(floor(w) * Float32(dY_46_u / t_4));
		end
		tmp_1 = tmp_2;
	elseif ((t_2 ^ Float32(2.0)) >= (t_0 ^ Float32(2.0)))
		tmp_1 = Float32(dX_46_u * Float32(floor(w) * t_5));
	else
		tmp_1 = Float32(floor(w) * Float32(dY_46_u * 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 = floor(h) * dY_46_v;
	t_1 = dX_46_u * floor(w);
	t_2 = dX_46_v * floor(h);
	t_3 = max((hypot(t_1, t_2) ^ single(2.0)), (hypot((floor(w) * dY_46_u), t_0) ^ single(2.0)));
	t_4 = sqrt(t_3);
	t_5 = sqrt((single(1.0) / t_3));
	tmp_2 = single(0.0);
	if (dX_46_v <= single(0.0010000000474974513))
		tmp_3 = single(0.0);
		if ((t_1 ^ single(2.0)) >= ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))
			tmp_3 = dX_46_u * (floor(w) / t_4);
		else
			tmp_3 = floor(w) * (dY_46_u / t_4);
		end
		tmp_2 = tmp_3;
	elseif ((t_2 ^ single(2.0)) >= (t_0 ^ single(2.0)))
		tmp_2 = dX_46_u * (floor(w) * t_5);
	else
		tmp_2 = floor(w) * (dY_46_u * t_5);
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.00100000005

   1. Initial program 81.2%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in w around 0 81.0%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in dY.u around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. unpow2 71.8%

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

      3. unpow2 71.8%

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

      4. swap-sqr 71.8%

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

      5. unpow2 71.8%

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

   10. Simplified 71.8%

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

   11. Taylor expanded in dX.u around inf 64.0%

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

       1. *-commutative 64.0%

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

       2. unpow2 64.0%

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

       3. unpow2 64.0%

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

       4. swap-sqr 64.0%

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

       5. unpow2 64.0%

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

       6. *-commutative 64.0%

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

   13. Simplified 64.0%

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

   14. Taylor expanded in dX.u around 0 64.2%

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

   16. Simplified 64.3%

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

   if 0.00100000005 < dX.v

   1. Initial program 75.0%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in w around 0 75.1%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in dY.u around 0 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. swap-sqr 63.5%

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

      5. unpow2 63.5%

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

   10. Simplified 63.5%

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

   11. Taylor expanded in dX.u around 0 62.3%

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

       1. unpow2 62.3%

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

       2. unpow2 62.3%

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

       3. swap-sqr 62.3%

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

       4. unpow2 62.3%

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

   13. Simplified 62.3%

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

Alternative 8: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := {t\_1}^{2}\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := dX.u \cdot \left\lfloor w\right\rfloor \\ t_5 := {\left(\mathsf{hypot}\left(t\_4, t\_0\right)\right)}^{2}\\ t_6 := \sqrt{\frac{1}{\mathsf{max}\left(t\_5, {\left(\mathsf{hypot}\left(t\_3, t\_1\right)\right)}^{2}\right)}}\\ t_7 := dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot t\_6\right)\\ \mathbf{if}\;dX.v \leq 0.0004349999944679439:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{t\_4}^{2} \geq t\_2:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot \frac{1}{\sqrt{\mathsf{max}\left(t\_5, {\left(\mathsf{hypot}\left(t\_1, t\_3\right)\right)}^{2}\right)}}\right)\\ \end{array}\\ \mathbf{elif}\;{t\_0}^{2} \geq t\_2:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot t\_6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
 :precision binary32
 (let* ((t_0 (* dX.v (floor h)))
        (t_1 (* (floor h) dY.v))
        (t_2 (pow t_1 2.0))
        (t_3 (* (floor w) dY.u))
        (t_4 (* dX.u (floor w)))
        (t_5 (pow (hypot t_4 t_0) 2.0))
        (t_6 (sqrt (/ 1.0 (fmax t_5 (pow (hypot t_3 t_1) 2.0)))))
        (t_7 (* dX.u (* (floor w) t_6))))
   (if (<= dX.v 0.0004349999944679439)
     (if (>= (pow t_4 2.0) t_2)
       t_7
       (*
        (floor w)
        (* dY.u (/ 1.0 (sqrt (fmax t_5 (pow (hypot t_1 t_3) 2.0)))))))
     (if (>= (pow t_0 2.0) t_2) t_7 (* (floor w) (* dY.u 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 = dX_46_v * floorf(h);
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = powf(t_1, 2.0f);
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = dX_46_u * floorf(w);
	float t_5 = powf(hypotf(t_4, t_0), 2.0f);
	float t_6 = sqrtf((1.0f / fmaxf(t_5, powf(hypotf(t_3, t_1), 2.0f))));
	float t_7 = dX_46_u * (floorf(w) * t_6);
	float tmp_1;
	if (dX_46_v <= 0.0004349999944679439f) {
		float tmp_2;
		if (powf(t_4, 2.0f) >= t_2) {
			tmp_2 = t_7;
		} else {
			tmp_2 = floorf(w) * (dY_46_u * (1.0f / sqrtf(fmaxf(t_5, powf(hypotf(t_1, t_3), 2.0f)))));
		}
		tmp_1 = tmp_2;
	} else if (powf(t_0, 2.0f) >= t_2) {
		tmp_1 = t_7;
	} else {
		tmp_1 = floorf(w) * (dY_46_u * 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(dX_46_v * floor(h))
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = t_1 ^ Float32(2.0)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = Float32(dX_46_u * floor(w))
	t_5 = hypot(t_4, t_0) ^ Float32(2.0)
	t_6 = sqrt(Float32(Float32(1.0) / ((t_5 != t_5) ? (hypot(t_3, t_1) ^ Float32(2.0)) : (((hypot(t_3, t_1) ^ Float32(2.0)) != (hypot(t_3, t_1) ^ Float32(2.0))) ? t_5 : max(t_5, (hypot(t_3, t_1) ^ Float32(2.0)))))))
	t_7 = Float32(dX_46_u * Float32(floor(w) * t_6))
	tmp_1 = Float32(0.0)
	if (dX_46_v <= Float32(0.0004349999944679439))
		tmp_2 = Float32(0.0)
		if ((t_4 ^ Float32(2.0)) >= t_2)
			tmp_2 = t_7;
		else
			tmp_2 = Float32(floor(w) * Float32(dY_46_u * Float32(Float32(1.0) / sqrt(((t_5 != t_5) ? (hypot(t_1, t_3) ^ Float32(2.0)) : (((hypot(t_1, t_3) ^ Float32(2.0)) != (hypot(t_1, t_3) ^ Float32(2.0))) ? t_5 : max(t_5, (hypot(t_1, t_3) ^ Float32(2.0)))))))));
		end
		tmp_1 = tmp_2;
	elseif ((t_0 ^ Float32(2.0)) >= t_2)
		tmp_1 = t_7;
	else
		tmp_1 = Float32(floor(w) * Float32(dY_46_u * 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 = dX_46_v * floor(h);
	t_1 = floor(h) * dY_46_v;
	t_2 = t_1 ^ single(2.0);
	t_3 = floor(w) * dY_46_u;
	t_4 = dX_46_u * floor(w);
	t_5 = hypot(t_4, t_0) ^ single(2.0);
	t_6 = sqrt((single(1.0) / max(t_5, (hypot(t_3, t_1) ^ single(2.0)))));
	t_7 = dX_46_u * (floor(w) * t_6);
	tmp_2 = single(0.0);
	if (dX_46_v <= single(0.0004349999944679439))
		tmp_3 = single(0.0);
		if ((t_4 ^ single(2.0)) >= t_2)
			tmp_3 = t_7;
		else
			tmp_3 = floor(w) * (dY_46_u * (single(1.0) / sqrt(max(t_5, (hypot(t_1, t_3) ^ single(2.0))))));
		end
		tmp_2 = tmp_3;
	elseif ((t_0 ^ single(2.0)) >= t_2)
		tmp_2 = t_7;
	else
		tmp_2 = floor(w) * (dY_46_u * t_6);
	end
	tmp_4 = tmp_2;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 4.34999994e-4

   1. Initial program 81.2%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in w around 0 81.0%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in dY.u around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. unpow2 71.8%

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

      3. unpow2 71.8%

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

      4. swap-sqr 71.8%

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

      5. unpow2 71.8%

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

   10. Simplified 71.8%

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

   11. Taylor expanded in dX.u around inf 64.0%

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

       1. *-commutative 64.0%

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

       2. unpow2 64.0%

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

       3. unpow2 64.0%

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

       4. swap-sqr 64.0%

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

       5. unpow2 64.0%

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

       6. *-commutative 64.0%

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

   13. Simplified 64.0%

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

   15. Applied egg-rr 64.1%

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

   if 4.34999994e-4 < dX.v

   1. Initial program 75.0%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in w around 0 75.1%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in dY.u around 0 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. swap-sqr 63.5%

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

      5. unpow2 63.5%

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

   10. Simplified 63.5%

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

   11. Taylor expanded in dX.u around 0 62.3%

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

       1. unpow2 62.3%

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

       2. unpow2 62.3%

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

       3. swap-sqr 62.3%

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

       4. unpow2 62.3%

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

   13. Simplified 62.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.0004349999944679439:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} \geq {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot \frac{1}{\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor h\right\rfloor \cdot dY.v, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}}\right)\\ \end{array}\\ \mathbf{elif}\;{\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} \geq {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}:\\ \;\;\;\;dX.u \cdot \left(\left\lfloor w\right\rfloor \cdot \sqrt{\frac{1}{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left\lfloor w\right\rfloor \cdot \left(dY.u \cdot \sqrt{\frac{1}{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.u \cdot \left\lfloor w\right\rfloor , dX.v \cdot \left\lfloor h\right\rfloor \right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.6%

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

5. Taylor expanded in w around 0 79.3%

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

7. Simplified 79.1%

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

8. Taylor expanded in dY.u around 0 69.5%

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

   1. *-commutative 69.5%

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

   2. unpow2 69.5%

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

   3. unpow2 69.5%

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

   4. swap-sqr 69.5%

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

   5. unpow2 69.5%

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

10. Simplified 69.5%

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

11. Taylor expanded in dX.u around inf 61.1%

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

    1. *-commutative 61.1%

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

    2. unpow2 61.1%

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

    3. unpow2 61.1%

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

    4. swap-sqr 61.1%

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

    5. unpow2 61.1%

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

    6. *-commutative 61.1%

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

13. Simplified 61.1%

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

15. Applied egg-rr 61.1%

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

16. Final simplification 61.1%

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

Alternative 10: 59.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.5%

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

3. Simplified 79.6%

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

5. Taylor expanded in w around 0 79.3%

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

7. Simplified 79.1%

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

8. Taylor expanded in dY.u around 0 69.5%

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

   1. *-commutative 69.5%

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

   2. unpow2 69.5%

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

   3. unpow2 69.5%

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

   4. swap-sqr 69.5%

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

   5. unpow2 69.5%

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

10. Simplified 69.5%

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

11. Taylor expanded in dX.u around inf 61.1%

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

    1. *-commutative 61.1%

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

    2. unpow2 61.1%

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

    3. unpow2 61.1%

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

    4. swap-sqr 61.1%

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

    5. unpow2 61.1%

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

    6. *-commutative 61.1%

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

13. Simplified 61.1%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, u)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dX.u)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor w) dY.u))))