UniformSampleCone 2

Percentage Accurate: 98.9% → 98.7%

Time: 20.7s

Alternatives: 6

Speedup: N/A×

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: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot yi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  ux
  (* (- 1.0 ux) (* maxCos zi))
  (*
   (sqrt (- 1.0 (* ux (* ux (* maxCos maxCos)))))
   (+
    (* (cos (* uy (* 2.0 PI))) xi)
    (* (sin (log1p (expm1 (* 2.0 (* uy PI))))) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(ux, ((1.0f - ux) * (maxCos * zi)), (sqrtf((1.0f - (ux * (ux * (maxCos * maxCos))))) * ((cosf((uy * (2.0f * ((float) M_PI)))) * xi) + (sinf(log1pf(expm1f((2.0f * (uy * ((float) M_PI)))))) * yi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(ux, Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * zi)), Float32(sqrt(Float32(Float32(1.0) - Float32(ux * Float32(ux * Float32(maxCos * maxCos))))) * Float32(Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * xi) + Float32(sin(log1p(expm1(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))) * yi))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

   1. associate-*r* 99.0%

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

   2. log1p-expm1-u 99.0%

      \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)\right)} \cdot yi\right)\right) \]

   3. *-commutative 99.0%

      \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(2 \cdot uy\right)} \cdot \pi\right)\right)\right) \cdot yi\right)\right) \]

   4. associate-*l* 99.0%

      \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{2 \cdot \left(uy \cdot \pi\right)}\right)\right)\right) \cdot yi\right)\right) \]

5. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \cdot yi\right)\right) \]

6. Taylor expanded in ux around 0 99.0%

   \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot \color{blue}{1}\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot yi\right)\right) \]

7. Final simplification 99.0%

   \[\leadsto \mathsf{fma}\left(ux, \left(1 - ux\right) \cdot \left(maxCos \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \cdot yi\right)\right) \]

Alternative 2: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in ux around 0 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 90.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in uy around 0 92.4%

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

   1. associate-*r* 92.4%

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

6. Simplified 92.4%

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

7. Final simplification 92.4%

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

Alternative 4: 90.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in uy around 0 92.4%

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

5. Taylor expanded in ux around 0 92.4%

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

6. Final simplification 92.4%

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

Alternative 5: 81.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(ux, maxCos \cdot \left(\left(1 - ux\right) \cdot zi\right), \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in uy around 0 92.4%

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

5. Taylor expanded in ux around 0 92.4%

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

6. Taylor expanded in uy around 0 82.7%

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

7. Taylor expanded in ux around 0 82.8%

   \[\leadsto \mathsf{fma}\left(ux, \color{blue}{-1 \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + maxCos \cdot zi}, \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot \left(maxCos \cdot 1\right)\right)\right)} \cdot \left(1 \cdot xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 82.8%

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

   2. *-lft-identity 82.8%

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

   3. mul-1-neg 82.8%

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

   4. *-commutative 82.8%

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

   5. associate-*l* 82.8%

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

   6. *-commutative 82.8%

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

   7. distribute-lft-neg-out 82.8%

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

   8. distribute-rgt-in 82.7%

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

   9. sub-neg 82.7%

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

   10. associate-*l* 82.8%

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

9. Simplified 82.8%

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

10. Final simplification 82.8%

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

Alternative 6: 78.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(ux, maxCos \cdot zi, \sqrt{1 - ux \cdot \left(ux \cdot \left(maxCos \cdot maxCos\right)\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in uy around 0 92.4%

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

5. Taylor expanded in ux around 0 92.4%

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

6. Taylor expanded in uy around 0 82.7%

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

7. Taylor expanded in ux around 0 79.6%

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

8. Final simplification 79.6%

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

Reproduce

herbie shell --seed 2023242 
(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)))