UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 23.6s

Alternatives: 23

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot \left(ux + -1\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right), 1\right)}\\ t_1 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathsf{fma}\left(1 - ux, zi \cdot \left(ux \cdot maxCos\right), \mathsf{fma}\left(t\_0 \cdot \sin t\_1, yi, xi \cdot \left(t\_0 \cdot \cos t\_1\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos (+ ux -1.0))
           (* (* ux ux) (* (- 1.0 ux) maxCos))
           1.0)))
        (t_1 (* 2.0 (* uy PI))))
   (fma
    (- 1.0 ux)
    (* zi (* ux maxCos))
    (fma (* t_0 (sin t_1)) yi (* xi (* t_0 (cos t_1)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * (ux + -1.0f)), ((ux * ux) * ((1.0f - ux) * maxCos)), 1.0f));
	float t_1 = 2.0f * (uy * ((float) M_PI));
	return fmaf((1.0f - ux), (zi * (ux * maxCos)), fmaf((t_0 * sinf(t_1)), yi, (xi * (t_0 * cosf(t_1)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * Float32(ux + Float32(-1.0))), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * maxCos)), Float32(1.0)))
	t_1 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return fma(Float32(Float32(1.0) - ux), Float32(zi * Float32(ux * maxCos)), fma(Float32(t_0 * sin(t_1)), yi, Float32(xi * Float32(t_0 * cos(t_1)))))
end

Derivation

1. Initial program 98.8%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(ux \cdot maxCos\right) \cdot zi, \mathsf{fma}\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(-\left(1 - ux\right) \cdot maxCos\right), 1\right)} \cdot xi, \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(-\left(1 - ux\right) \cdot maxCos\right), 1\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\right)\right)} \]

6. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(ux \cdot maxCos\right) \cdot zi, \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot \left(-1 + ux\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot \left(-1 + ux\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right), 1\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)}\right) \]

7. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, zi \cdot \left(ux \cdot maxCos\right), \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot \left(ux + -1\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right), 1\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), yi, xi \cdot \left(\sqrt{\mathsf{fma}\left(maxCos \cdot \left(ux + -1\right), \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right), 1\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{\mathsf{fma}\left(xi, \cos t\_0, \sin t\_0 \cdot yi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}}{zi}\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (*
    zi
    (fma
     maxCos
     (* ux (- 1.0 ux))
     (/
      (*
       (fma xi (cos t_0) (* (sin t_0) yi))
       (sqrt
        (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0)))
      zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return zi * fmaf(maxCos, (ux * (1.0f - ux)), ((fmaf(xi, cosf(t_0), (sinf(t_0) * yi)) * sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f))) / zi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return Float32(zi * fma(maxCos, Float32(ux * Float32(Float32(1.0) - ux)), Float32(Float32(fma(xi, cos(t_0), Float32(sin(t_0) * yi)) * sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))) / zi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(zi \cdot \left(-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right) + -1 \cdot \frac{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)} + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}}{zi}\right)\right)} \]

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{-zi \cdot \left(-\mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}{zi}\right)\right)} \]

6. Final simplification 98.3%

   \[\leadsto zi \cdot \mathsf{fma}\left(maxCos, ux \cdot \left(1 - ux\right), \frac{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}}{zi}\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))