Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.1%

Time: 14.6s

Alternatives: 7

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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (+
   0.5
   (-
    (* 0.5 (cos (* u2 (* (pow (cbrt PI) 3.0) 2.0))))
    (pow (sin (* u2 PI)) 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (0.5f + ((0.5f * cosf((u2 * (powf(cbrtf(((float) M_PI)), 3.0f) * 2.0f)))) - powf(sinf((u2 * ((float) M_PI))), 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(0.5) + Float32(Float32(Float32(0.5) * cos(Float32(u2 * Float32((cbrt(Float32(pi)) ^ Float32(3.0)) * Float32(2.0))))) - (sin(Float32(u2 * Float32(pi))) ^ Float32(2.0)))))
end

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow2 98.6%

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

   3. associate-*l* 98.6%

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

5. Applied egg-rr 98.6%

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

   1. unpow2 98.6%

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

   2. add-sqr-sqrt 98.8%

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

   3. cos-2 98.6%

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

   4. cancel-sign-sub-inv 98.6%

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

   5. sqr-cos-a 98.6%

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

   6. associate-+l+ 98.7%

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

   7. associate-*r* 98.7%

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

   8. *-commutative 98.7%

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

   9. associate-*l* 98.7%

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

7. Applied egg-rr 98.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \left(-\sin \left(\pi \cdot u2\right)\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 98.7%

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

   2. unpow2 98.7%

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

9. Simplified 98.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. add-cube-cbrt 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

    2. pow3 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

11. Applied egg-rr 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

12. Final simplification 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) - {\sin \left(u2 \cdot {\left(\sqrt[3]{\pi}\right)}^{3}\right)}^{2}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (+
   0.5
   (-
    (* 0.5 (cos (* u2 (* PI 2.0))))
    (pow (sin (* u2 (pow (cbrt PI) 3.0))) 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (0.5f + ((0.5f * cosf((u2 * (((float) M_PI) * 2.0f)))) - powf(sinf((u2 * powf(cbrtf(((float) M_PI)), 3.0f))), 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(0.5) + Float32(Float32(Float32(0.5) * cos(Float32(u2 * Float32(Float32(pi) * Float32(2.0))))) - (sin(Float32(u2 * (cbrt(Float32(pi)) ^ Float32(3.0)))) ^ Float32(2.0)))))
end

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.6%

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

   2. pow2 98.6%

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

   3. associate-*l* 98.6%

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

5. Applied egg-rr 98.6%

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

   1. unpow2 98.6%

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

   2. add-sqr-sqrt 98.8%

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

   3. cos-2 98.6%

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

   4. cancel-sign-sub-inv 98.6%

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

   5. sqr-cos-a 98.6%

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

   6. associate-+l+ 98.7%

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

   7. associate-*r* 98.7%

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

   8. *-commutative 98.7%

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

   9. associate-*l* 98.7%

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

7. Applied egg-rr 98.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \left(-\sin \left(\pi \cdot u2\right)\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right)} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 98.7%

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

   2. unpow2 98.7%

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

9. Simplified 98.7%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. add-cube-cbrt 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

    2. pow3 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot 2\right)\right) - {\sin \left(u2 \cdot \pi\right)}^{2}\right)\right) \]

11. Applied egg-rr 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) - {\sin \left(u2 \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}\right)}^{2}\right)\right) \]

12. Final simplification 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(0.5 + \left(0.5 \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) - {\sin \left(u2 \cdot {\left(\sqrt[3]{\pi}\right)}^{3}\right)}^{2}\right)\right) \]

Alternative 3: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

   1. add-log-exp 98.8%

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

   2. *-commutative 98.8%

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

   3. exp-prod 98.4%

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

5. Applied egg-rr 98.4%

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

   1. pow-exp 98.8%

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

   2. *-commutative 98.8%

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

   3. *-commutative 98.8%

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

   4. associate-*l* 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.009999999776482582)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.009999999776482582f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.009999999776482582))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00999999978

   1. Initial program 57.3%

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

      1. sub-neg 57.3%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in u2 around 0 97.0%

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

   if 0.00999999978 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 58.7%

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

   2. Taylor expanded in u1 around 0 74.7%

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

      1. mul-1-neg 74.7%

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

   4. Simplified 74.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 6: 80.6% 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 57.7%

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

   1. sub-neg 57.7%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u2 around 0 77.1%

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

5. Final simplification 77.1%

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

Alternative 7: 65.6% 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 57.7%

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

   1. sub-neg 57.7%

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

   2. log1p-def 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u2 around 0 77.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 76.4%

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

   2. pow2 76.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{-\mathsf{log1p}\left(-u1\right)}}\right)}^{2}} \cdot 1 \]

   3. pow1/2 76.4%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}}}\right)}^{2} \cdot 1 \]

   4. sqrt-pow1 76.4%

      \[\leadsto {\color{blue}{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot 1 \]

   5. add-sqr-sqrt 76.4%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   6. sqrt-unprod 76.4%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{\left(-\mathsf{log1p}\left(-u1\right)\right) \cdot \left(-\mathsf{log1p}\left(-u1\right)\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   7. sqr-neg 76.4%

      \[\leadsto {\left({\left(\sqrt{\color{blue}{\mathsf{log1p}\left(-u1\right) \cdot \mathsf{log1p}\left(-u1\right)}}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   8. sqrt-unprod -0.0%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{\mathsf{log1p}\left(-u1\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   9. add-sqr-sqrt -0.0%

      \[\leadsto {\left({\color{blue}{\left(\mathsf{log1p}\left(-u1\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   10. add-sqr-sqrt -0.0%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{-u1} \cdot \sqrt{-u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   11. sqrt-unprod 61.3%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   12. sqr-neg 61.3%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(\sqrt{\color{blue}{u1 \cdot u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   13. sqrt-unprod 61.3%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   14. add-sqr-sqrt 61.3%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{u1}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot 1 \]

   15. metadata-eval 61.3%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot 1 \]

6. Applied egg-rr 61.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot 1 \]

7. Taylor expanded in u1 around 0 62.9%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]

8. Final simplification 62.9%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023301 
(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))))