Anisotropic x16 LOD (line direction, v)

Percentage Accurate: 76.3% → 76.4%

Time: 23.0s

Alternatives: 8

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 76.4% accurate, N/A× speedup

Math

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

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.7%

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

6. Applied rewrites 77.8%

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

7. Final simplification 77.8%

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

Alternative 2: 76.3% accurate, N/A× speedup

Math

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

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

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

Julia

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.7%

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

6. Applied rewrites 77.8%

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

8. Applied rewrites 77.8%

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

9. Final simplification 77.8%

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

Alternative 3: 76.3% accurate, N/A× speedup

Math

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

FPCore

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

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

Julia

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.7%

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

6. Applied rewrites 77.8%

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

8. Applied rewrites 77.8%

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

9. Final simplification 77.8%

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

Alternative 4: 76.3% accurate, N/A× speedup

Math

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

FPCore

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

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 = -1.0f * (floorf(w) * dY_46_u);
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = -1.0f * (floorf(w) * dX_46_u);
	float t_4 = (t_3 * t_3) + (t_0 * t_0);
	float t_5 = floorf(h) * dY_46_v;
	float t_6 = (t_1 * t_1) + (t_5 * t_5);
	float t_7 = powf(floorf(w), 2.0f);
	float tmp;
	if (t_4 >= t_6) {
		tmp = (1.0f / sqrtf(fmaxf(t_4, t_6))) * t_0;
	} else {
		tmp = (powf(fmaxf((((t_2 * dX_46_v) * dX_46_v) - ((-1.0f * (dX_46_u * t_7)) * dX_46_u)), (((t_2 * dY_46_v) * dY_46_v) - ((-1.0f * (dY_46_u * t_7)) * dY_46_u))), -0.5f) * dY_46_v) * floorf(h);
	}
	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(Float32(-1.0) * Float32(floor(w) * dY_46_u))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = Float32(Float32(-1.0) * Float32(floor(w) * dX_46_u))
	t_4 = Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0))
	t_5 = Float32(floor(h) * dY_46_v)
	t_6 = Float32(Float32(t_1 * t_1) + Float32(t_5 * t_5))
	t_7 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (t_4 >= t_6)
		tmp = Float32(Float32(Float32(1.0) / sqrt(fmax(t_4, t_6))) * t_0);
	else
		tmp = Float32(Float32((fmax(Float32(Float32(Float32(t_2 * dX_46_v) * dX_46_v) - Float32(Float32(Float32(-1.0) * Float32(dX_46_u * t_7)) * dX_46_u)), Float32(Float32(Float32(t_2 * dY_46_v) * dY_46_v) - Float32(Float32(Float32(-1.0) * Float32(dY_46_u * t_7)) * dY_46_u))) ^ Float32(-0.5)) * dY_46_v) * floor(h));
	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 = single(-1.0) * (floor(w) * dY_46_u);
	t_2 = floor(h) ^ single(2.0);
	t_3 = single(-1.0) * (floor(w) * dX_46_u);
	t_4 = (t_3 * t_3) + (t_0 * t_0);
	t_5 = floor(h) * dY_46_v;
	t_6 = (t_1 * t_1) + (t_5 * t_5);
	t_7 = floor(w) ^ single(2.0);
	tmp = single(0.0);
	if (t_4 >= t_6)
		tmp = (single(1.0) / sqrt(max(t_4, t_6))) * t_0;
	else
		tmp = ((max((((t_2 * dX_46_v) * dX_46_v) - ((single(-1.0) * (dX_46_u * t_7)) * dX_46_u)), (((t_2 * dY_46_v) * dY_46_v) - ((single(-1.0) * (dY_46_u * t_7)) * dY_46_u))) ^ single(-0.5)) * dY_46_v) * floor(h);
	end
	tmp_2 = tmp;
end

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.7%

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

5. Final simplification 77.7%

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

Alternative 5: 76.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.7%

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

6. Applied rewrites 77.7%

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

7. Final simplification 77.7%

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

Alternative 6: 76.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.7%

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

6. Applied rewrites 77.7%

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

7. Final simplification 77.7%

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

Alternative 7: 76.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.6%

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

6. Applied rewrites 77.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-floor.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. mul-1-neg N/A

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

   9. associate-*l* N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-pow.f32 N/A

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

   14. lift-floor.f32 N/A

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

   15. mul-1-neg N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f32 77.5

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

8. Applied rewrites 77.5%

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

9. Final simplification 77.5%

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

Alternative 8: 76.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.6%

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

4. Applied rewrites 77.6%

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

6. Applied rewrites 77.5%

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

7. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2025064 
(FPCore (w h dX.u dX.v dY.u dY.v maxAniso)
  :name "Anisotropic x16 LOD (line direction, v)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (== maxAniso 16.0))
  (if (>= (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dX.v)) (* (/ 1.0 (sqrt (fmax (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v)))))) (* (floor h) dY.v))))