UniformSampleCone, x

Percentage Accurate: 57.9% → 58.0%

Time: 1.2min

Alternatives: 4

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 58.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 (* (log E) PI))))
  (sqrt (fma (fma ux maxCos (- 1.0 ux)) (- -1.0 (* ux (+ maxCos -1.0))) 1.0))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(log(Float32(exp(1))) * Float32(pi))))) * sqrt(fma(fma(ux, maxCos, Float32(Float32(1.0) - ux)), Float32(Float32(-1.0) - Float32(ux * Float32(maxCos + Float32(-1.0)))), Float32(1.0))))
end

Derivation

1. Initial program 55.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.5%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. sub-neg 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   3. +-commutative 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + 1}} \]

   4. distribute-rgt-neg-in 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

   5. fma-def 55.5%

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

3. Simplified 55.5%

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

   1. log1p-expm1-u 55.5%

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

   2. expm1-udef 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(\color{blue}{e^{\pi} - 1}\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   3. *-un-lft-identity 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(e^{\color{blue}{1 \cdot \pi}} - 1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   4. exp-prod 55.6%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(\color{blue}{{\left(e^{1}\right)}^{\pi}} - 1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   5. pow-to-exp 55.6%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{1}\right) \cdot \pi}} - 1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   6. expm1-def 55.6%

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

   7. log1p-expm1-u 55.6%

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

   8. exp-1-e 55.6%

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

6. Applied egg-rr 55.6%

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

7. Final simplification 55.6%

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

Alternative 2: 58.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. associate-*l* 55.5%

      \[\leadsto \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. sub-neg 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   3. +-commutative 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + 1}} \]

   4. distribute-rgt-neg-in 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

   5. fma-def 55.5%

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

3. Simplified 55.5%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)}} \]
4. Add Preprocessing

5. Final simplification 55.5%

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

Alternative 3: 57.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(\log e \cdot \pi\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* (log E) PI) (* uy 2.0))) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((logf(((float) M_E)) * ((float) M_PI)) * (uy * 2.0f))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(log(Float32(exp(1))) * Float32(pi)) * Float32(uy * Float32(2.0)))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((log(single(2.71828182845904523536)) * single(pi)) * (uy * single(2.0)))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Derivation

1. Initial program 55.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. log1p-expm1-u 55.5%

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

   2. expm1-udef 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(\color{blue}{e^{\pi} - 1}\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   3. *-un-lft-identity 55.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(e^{\color{blue}{1 \cdot \pi}} - 1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   4. exp-prod 55.6%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(\color{blue}{{\left(e^{1}\right)}^{\pi}} - 1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   5. pow-to-exp 55.6%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{1}\right) \cdot \pi}} - 1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), -1 - ux \cdot \left(maxCos + -1\right), 1\right)} \]

   6. expm1-def 55.6%

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

   7. log1p-expm1-u 55.6%

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

   8. exp-1-e 55.6%

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

4. Applied egg-rr 55.5%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \color{blue}{\left(\log e \cdot \pi\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Final simplification 55.5%

   \[\leadsto \cos \left(\left(\log e \cdot \pi\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Add Preprocessing

Alternative 4: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sqrt{1 - t\_0 \cdot t\_0} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sqrt (- 1.0 (* t_0 t_0))) (cos (* PI (* uy 2.0))))))

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = sqrt((single(1.0) - (t_0 * t_0))) * cos((single(pi) * (uy * single(2.0))));
end

Derivation

1. Initial program 55.5%

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

3. Final simplification 55.5%

   \[\leadsto \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (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) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))