Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.0%

Time: 14.0s

Alternatives: 8

Speedup: 4.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\sqrt[3]{{\left(2 \cdot \pi\right)}^{3} \cdot {u2}^{3}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (cos (cbrt (* (pow (* 2.0 PI) 3.0) (pow u2 3.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf(cbrtf((powf((2.0f * ((float) M_PI)), 3.0f) * powf(u2, 3.0f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(cbrt(Float32((Float32(Float32(2.0) * Float32(pi)) ^ Float32(3.0)) * (u2 ^ Float32(3.0))))))
end

Derivation

1. Initial program 54.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 54.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Step-by-step derivation

   1. add-cbrt-cube 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\color{blue}{\sqrt[3]{\left(\left(2 \cdot \pi\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(2 \cdot \pi\right)}} \cdot u2\right) \]

   2. add-cbrt-cube 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\sqrt[3]{\left(\left(2 \cdot \pi\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(2 \cdot \pi\right)} \cdot \color{blue}{\sqrt[3]{\left(u2 \cdot u2\right) \cdot u2}}\right) \]

   3. cbrt-unprod 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\sqrt[3]{\left(\left(\left(2 \cdot \pi\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right)} \]

   4. pow3 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\sqrt[3]{\color{blue}{{\left(2 \cdot \pi\right)}^{3}} \cdot \left(\left(u2 \cdot u2\right) \cdot u2\right)}\right) \]

   5. pow3 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\sqrt[3]{{\left(2 \cdot \pi\right)}^{3} \cdot \color{blue}{{u2}^{3}}}\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\sqrt[3]{{\left(2 \cdot \pi\right)}^{3} \cdot {u2}^{3}}\right)} \]

6. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\sqrt[3]{{\left(2 \cdot \pi\right)}^{3} \cdot {u2}^{3}}\right) \]

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(e^{\log \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (exp (log (* 2.0 (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf(expf(logf((2.0f * (((float) M_PI) * u2)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(exp(log(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))))
end

Derivation

1. Initial program 54.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 54.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Step-by-step derivation

   1. add-exp-log 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(e^{\log \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} \]

   2. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(e^{\log \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}}\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(e^{\log \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]

6. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(e^{\log \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \]

Alternative 3: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.9999719858169556:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999719858169556)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9999719858169556f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999719858169556))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.999971986

   1. Initial program 52.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 52.6%

         \[\leadsto \color{blue}{\left(\sqrt{\sqrt{-\log \left(1 - u1\right)}} \cdot \sqrt{\sqrt{-\log \left(1 - u1\right)}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. pow2 52.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{-\log \left(1 - u1\right)}}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. pow1/2 52.6%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(-\log \left(1 - u1\right)\right)}^{0.5}}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. sqrt-pow1 52.7%

         \[\leadsto {\color{blue}{\left({\left(-\log \left(1 - u1\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. add-sqr-sqrt 52.6%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. sqrt-unprod 52.7%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\left(-\log \left(1 - u1\right)\right) \cdot \left(-\log \left(1 - u1\right)\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. sqr-neg 52.7%

         \[\leadsto {\left({\left(\sqrt{\color{blue}{\log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. sqrt-unprod 2.0%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{\log \left(1 - u1\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. add-sqr-sqrt 2.0%

         \[\leadsto {\left({\color{blue}{\log \left(1 - u1\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. sub-neg 2.0%

          \[\leadsto {\left({\log \color{blue}{\left(1 + \left(-u1\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. log1p-udef -0.0%

          \[\leadsto {\left({\color{blue}{\left(\mathsf{log1p}\left(-u1\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{-u1} \cdot \sqrt{-u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. sqrt-unprod 76.2%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. sqr-neg 76.2%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\sqrt{\color{blue}{u1 \cdot u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. sqrt-unprod 76.2%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. add-sqr-sqrt 76.2%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{u1}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. metadata-eval 76.2%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u1 around 0 78.2%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.999971986 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 55.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 55.4%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-def 99.5%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

   4. Taylor expanded in u2 around 0 96.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999719858169556:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.0007650000043213367:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0007650000043213367)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (* u1 (- (- -1.0) (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0007650000043213367f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (-(-1.0f) - (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0007650000043213367))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(-Float32(-1.0)) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 7.65000004e-4

   1. Initial program 55.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 55.2%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-def 99.6%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

   4. Taylor expanded in u2 around 0 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

   if 7.65000004e-4 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 53.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0 89.5%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative 89.5%

         \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot -1} + -0.5 \cdot {u1}^{2}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. *-commutative 89.5%

         \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{{u1}^{2} \cdot -0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. unpow2 89.5%

         \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. associate-*l* 89.5%

         \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-lft-out 89.5%

         \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Simplified 89.5%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0007650000043213367:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 54.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 54.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

4. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

Alternative 6: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

Derivation

1. Initial program 54.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 54.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

4. Taylor expanded in u2 around 0 79.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

5. Final simplification 79.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

Alternative 7: 72.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (- (- -1.0) (* u1 -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (-(-1.0f) - (u1 * -0.5f))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 * (-(-1.0e0) - (u1 * (-0.5e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(-Float32(-1.0)) - Float32(u1 * Float32(-0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (-single(-1.0) - (u1 * single(-0.5)))));
end

Derivation

1. Initial program 54.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u1 around 0 87.7%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Step-by-step derivation

   1. *-commutative 87.7%

      \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot -1} + -0.5 \cdot {u1}^{2}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. *-commutative 87.7%

      \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{{u1}^{2} \cdot -0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. unpow2 87.7%

      \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. associate-*l* 87.7%

      \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. distribute-lft-out 87.6%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Simplified 87.6%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0 71.5%

   \[\leadsto \sqrt{-u1 \cdot \left(-1 + u1 \cdot -0.5\right)} \cdot \color{blue}{1} \]

6. Final simplification 71.5%

   \[\leadsto \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)} \]

Alternative 8: 64.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 54.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 54.5%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

4. Taylor expanded in u2 around 0 79.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

5. Taylor expanded in u1 around 0 65.0%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
6. Step-by-step derivation

   1. mul-1-neg 65.0%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot 1 \]

7. Simplified 65.0%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot 1 \]

8. Final simplification 65.0%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023334 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))