Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.3%

Time: 13.8s

Alternatives: 17

Speedup: 8.9×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(\frac{-1}{\frac{1}{u1}}\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (/ -1.0 (/ 1.0 u1))))) (sin (* PI (+ u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf((-1.0f / (1.0f / u1)))) * sinf((((float) M_PI) * (u2 + u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(Float32(-1.0) / Float32(Float32(1.0) / u1))))) * sin(Float32(Float32(pi) * Float32(u2 + u2))))
end

Derivation

1. Initial program 56.1%

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

   1. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. neg-lowering-neg.f32 98.4

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

4. Applied egg-rr 98.4%

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

   1. associate-*l* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   2. count-2 N/A

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

   3. sin-lowering-sin.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. *-lowering-*.f32 N/A

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

   6. PI-lowering-PI.f32 N/A

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

   7. +-lowering-+.f32 98.4

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

6. Applied egg-rr 98.4%

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

   1. neg-sub0 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{0 - u1}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   2. flip3-- N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\frac{{0}^{3} - {u1}^{3}}{0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   3. frac-2neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} - {u1}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\left(\color{blue}{0} - {u1}^{3}\right)\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({u1}^{3}\right)\right)}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   6. cube-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   7. sqr-pow N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)}}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   8. pow-prod-down N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{{\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)\right)}^{\left(\frac{3}{2}\right)}}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   9. sqr-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left({\color{blue}{\left(u1 \cdot u1\right)}}^{\left(\frac{3}{2}\right)}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   10. pow-prod-down N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{{u1}^{\left(\frac{3}{2}\right)} \cdot {u1}^{\left(\frac{3}{2}\right)}}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   11. sqr-pow N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\mathsf{neg}\left(\color{blue}{{u1}^{3}}\right)}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   12. cube-neg N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   13. sqr-pow N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(u1\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   14. pow-prod-down N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   15. sqr-neg N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{{\color{blue}{\left(u1 \cdot u1\right)}}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   16. pow-prod-down N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\color{blue}{{u1}^{\left(\frac{3}{2}\right)} \cdot {u1}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   17. sqr-pow N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{\color{blue}{{u1}^{3}}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\frac{{u1}^{3}}{\mathsf{neg}\left(\left(0 \cdot 0 + \left(u1 \cdot u1 + 0 \cdot u1\right)\right)\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

8. Applied egg-rr 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{\frac{u1 \cdot \left(u1 \cdot u1\right)}{-\mathsf{fma}\left(u1, u1, 0\right)}}\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(\left(u1 \cdot u1 + 0\right)\right)}{u1 \cdot \left(u1 \cdot u1\right)}}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(\left(u1 \cdot u1 + 0\right)\right)}{u1 \cdot \left(u1 \cdot u1\right)}}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   3. clear-num N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{1}{\frac{u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(u1 \cdot u1 + 0\right)\right)}}}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{1}{\frac{u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\left(u1 \cdot u1 + 0\right)\right)}}}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   5. +-rgt-identity N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\frac{u1 \cdot \left(u1 \cdot u1\right)}{\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)}}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{u1 \cdot \left(u1 \cdot u1\right)}{u1 \cdot u1}\right)}}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   7. cube-unmult N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\frac{\color{blue}{{u1}^{3}}}{u1 \cdot u1}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   8. pow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\frac{{u1}^{3}}{\color{blue}{{u1}^{2}}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   9. pow-div N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\color{blue}{{u1}^{\left(3 - 2\right)}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left({u1}^{\color{blue}{1}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left({u1}^{\color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   12. pow-prod-up N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\color{blue}{{u1}^{\frac{1}{2}} \cdot {u1}^{\frac{1}{2}}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   13. pow1/2 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\color{blue}{\sqrt{u1}} \cdot {u1}^{\frac{1}{2}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   14. pow1/2 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\sqrt{u1} \cdot \color{blue}{\sqrt{u1}}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   15. rem-square-sqrt N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{neg}\left(\color{blue}{u1}\right)}}\right)\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

   16. neg-lowering-neg.f32 98.5

       \[\leadsto \sqrt{-\mathsf{log1p}\left(\frac{1}{\frac{1}{\color{blue}{-u1}}}\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]

10. Applied egg-rr 98.5%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{1}{-u1}}}\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]

11. Final simplification 98.5%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(\frac{-1}{\frac{1}{u1}}\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]
12. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* PI (+ u2 u2))) (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((((float) M_PI) * (u2 + u2))) * sqrtf(-log1pf(-u1));
}

Julia

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

Derivation

1. Initial program 56.1%

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

   1. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. accelerator-lowering-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. neg-lowering-neg.f32 98.4

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

4. Applied egg-rr 98.4%

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

   1. associate-*l* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   2. count-2 N/A

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

   3. sin-lowering-sin.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. *-lowering-*.f32 N/A

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

   6. PI-lowering-PI.f32 N/A

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

   7. +-lowering-+.f32 98.4

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

   \[\leadsto \sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
8. Add Preprocessing

Alternative 3: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.125, \sqrt{u1} \cdot 0.13541666666666666\right), \sqrt{u1} \cdot 0.25\right), \sqrt{u1}\right) \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.09000000357627869)
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* PI 2.0))))
     (*
      (fma
       u1
       (fma
        u1
        (fma (sqrt (* u1 (* u1 u1))) 0.125 (* (sqrt u1) 0.13541666666666666))
        (* (sqrt u1) 0.25))
       (sqrt u1))
      (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.09000000357627869f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (((float) M_PI) * 2.0f)));
	} else {
		tmp = fmaf(u1, fmaf(u1, fmaf(sqrtf((u1 * (u1 * u1))), 0.125f, (sqrtf(u1) * 0.13541666666666666f)), (sqrtf(u1) * 0.25f)), sqrtf(u1)) * sinf(t_0);
	}
	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(0.09000000357627869))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(pi) * Float32(2.0)))));
	else
		tmp = Float32(fma(u1, fma(u1, fma(sqrt(Float32(u1 * Float32(u1 * u1))), Float32(0.125), Float32(sqrt(u1) * Float32(0.13541666666666666))), Float32(sqrt(u1) * Float32(0.25))), sqrt(u1)) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0900000036

   1. Initial program 56.3%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      11. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      12. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      14. PI-lowering-PI.f32 98.6

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

   if 0.0900000036 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 48.8%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right), \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.16666666666666666, \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(0.25 + \frac{-0.0625}{u1}\right)\right), \sqrt{\frac{1}{u1}} \cdot 0.25\right), \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\left(\sqrt{u1} + u1 \cdot \left(\frac{1}{4} \cdot \sqrt{u1} + u1 \cdot \left(\frac{-1}{32} \cdot \sqrt{u1} + \left(\frac{1}{8} \cdot \sqrt{{u1}^{3}} + \frac{1}{6} \cdot \sqrt{u1}\right)\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(u1 \cdot \left(\frac{1}{4} \cdot \sqrt{u1} + u1 \cdot \left(\frac{-1}{32} \cdot \sqrt{u1} + \left(\frac{1}{8} \cdot \sqrt{{u1}^{3}} + \frac{1}{6} \cdot \sqrt{u1}\right)\right)\right) + \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(u1, \frac{1}{4} \cdot \sqrt{u1} + u1 \cdot \left(\frac{-1}{32} \cdot \sqrt{u1} + \left(\frac{1}{8} \cdot \sqrt{{u1}^{3}} + \frac{1}{6} \cdot \sqrt{u1}\right)\right), \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Simplified 91.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.125, \sqrt{u1} \cdot 0.13541666666666666\right), \sqrt{u1} \cdot 0.25\right), \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.125, \sqrt{u1} \cdot 0.13541666666666666\right), \sqrt{u1} \cdot 0.25\right), \sqrt{u1}\right) \cdot \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.09000000357627869)
   (*
    (sqrt (- (log1p (- u1))))
    (* u2 (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* PI 2.0))))
   (*
    (sin (* PI (+ u2 u2)))
    (sqrt (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.09000000357627869f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (((float) M_PI) * 2.0f)));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.09000000357627869))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(pi) * Float32(2.0)))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0900000036

   1. Initial program 56.3%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      11. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      12. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      14. PI-lowering-PI.f32 98.6

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right) \]

   7. Simplified 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

   if 0.0900000036 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.2%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 97.4

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

   4. Applied egg-rr 97.4%

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

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      2. count-2 N/A

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

      3. sin-lowering-sin.f32 N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. +-lowering-+.f32 97.4

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

   6. Applied egg-rr 97.4%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, \frac{1}{3} + \frac{1}{4} \cdot u1, \frac{1}{2}\right)}, 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, \frac{1}{2}\right), 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      8. accelerator-lowering-fma.f32 91.7

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 91.7%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.003000000026077032)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sin (* PI (+ u2 u2)))
    (sqrt (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 97.9

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      2. count-2 N/A

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

      3. sin-lowering-sin.f32 N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. +-lowering-+.f32 97.9

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, \frac{1}{3} + \frac{1}{4} \cdot u1, \frac{1}{2}\right)}, 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, \frac{1}{2}\right), 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      8. accelerator-lowering-fma.f32 93.9

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right)}, 0.5\right), 1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 93.9%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.003000000026077032)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sin (* PI (+ u2 u2)))
    (sqrt (* u1 (fma u1 (fma u1 0.3333333333333333 0.5) 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf((u1 * fmaf(u1, fmaf(u1, 0.3333333333333333f, 0.5f), 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(u1 * fma(u1, fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 97.9

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      2. count-2 N/A

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

      3. sin-lowering-sin.f32 N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. +-lowering-+.f32 97.9

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) + 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{1}{2} + \frac{1}{3} \cdot u1, 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{1}{3} \cdot u1 + \frac{1}{2}}, 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{1}{3}} + \frac{1}{2}, 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. accelerator-lowering-fma.f32 92.2

         \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right)}, 1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 92.2%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.003000000026077032)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sin (* PI (+ u2 u2))) (sqrt (* u1 (fma u1 0.5 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf((u1 * fmaf(u1, 0.5f, 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 97.9

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      2. count-2 N/A

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

      3. sin-lowering-sin.f32 N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. +-lowering-+.f32 97.9

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(\frac{1}{2} \cdot u1 + 1\right)}} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \left(\color{blue}{u1 \cdot \frac{1}{2}} + 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. accelerator-lowering-fma.f32 89.2

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)}} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 89.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.00800000037997961)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sin (* PI (+ u2 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.00800000037997961f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.00800000037997961))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

      1. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(2 \cdot \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. PI-lowering-PI.f32 97.2

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

   7. Simplified 97.2%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      4. count-2 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. sin-lowering-sin.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(u2 + u2\right)\right) \]

      9. +-lowering-+.f32 79.6

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.00800000037997961:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{u1}, 0.5 \cdot \left(0.25 + \frac{-0.0625}{u1}\right), t\_0 \cdot 0.16666666666666666\right), 0.25 \cdot t\_0\right), \sqrt{u1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))))
   (if (<= (* u2 (* PI 2.0)) 0.00800000037997961)
     (*
      2.0
      (*
       u2
       (*
        PI
        (fma
         (* u1 u1)
         (fma
          u1
          (fma
           (sqrt u1)
           (* 0.5 (+ 0.25 (/ -0.0625 u1)))
           (* t_0 0.16666666666666666))
          (* 0.25 t_0))
         (sqrt u1)))))
     (* (sin (* PI (+ u2 u2))) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((1.0f / u1));
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.00800000037997961f) {
		tmp = 2.0f * (u2 * (((float) M_PI) * fmaf((u1 * u1), fmaf(u1, fmaf(sqrtf(u1), (0.5f * (0.25f + (-0.0625f / u1))), (t_0 * 0.16666666666666666f)), (0.25f * t_0)), sqrtf(u1))));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(Float32(1.0) / u1))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.00800000037997961))
		tmp = Float32(Float32(2.0) * Float32(u2 * Float32(Float32(pi) * fma(Float32(u1 * u1), fma(u1, fma(sqrt(u1), Float32(Float32(0.5) * Float32(Float32(0.25) + Float32(Float32(-0.0625) / u1))), Float32(t_0 * Float32(0.16666666666666666))), Float32(Float32(0.25) * t_0)), sqrt(u1)))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 51.2%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right), \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.16666666666666666, \left(0.5 \cdot \sqrt{u1}\right) \cdot \left(0.25 + \frac{-0.0625}{u1}\right)\right), \sqrt{\frac{1}{u1}} \cdot 0.25\right), \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right)} \]

      3. *-lowering-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)}\right) \]

      4. PI-lowering-PI.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \sqrt{u1}\right)}\right)\right) \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right), \sqrt{u1}\right)}\right)\right) \]

   10. Simplified 91.5%

       \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{u1}, \left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5, \sqrt{\frac{1}{u1}} \cdot 0.16666666666666666\right), \sqrt{\frac{1}{u1}} \cdot 0.25\right), \sqrt{u1}\right)\right)\right)} \]

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      4. count-2 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. sin-lowering-sin.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(u2 + u2\right)\right) \]

      9. +-lowering-+.f32 79.6

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.00800000037997961:\\ \;\;\;\;2 \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(\sqrt{u1}, 0.5 \cdot \left(0.25 + \frac{-0.0625}{u1}\right), \sqrt{\frac{1}{u1}} \cdot 0.16666666666666666\right), 0.25 \cdot \sqrt{\frac{1}{u1}}\right), \sqrt{u1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.15000000596046448:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 2\right)\right) \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.15000000596046448)
   (*
    (* u2 (fma u2 (* u2 (* PI (* -1.3333333333333333 (* PI PI)))) (* PI 2.0)))
    (fma (sqrt (* u1 (* u1 u1))) 0.25 (sqrt u1)))
   (* (sin (* PI (+ u2 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.15000000596046448f) {
		tmp = (u2 * fmaf(u2, (u2 * (((float) M_PI) * (-1.3333333333333333f * (((float) M_PI) * ((float) M_PI))))), (((float) M_PI) * 2.0f))) * fmaf(sqrtf((u1 * (u1 * u1))), 0.25f, sqrtf(u1));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.15000000596046448))
		tmp = Float32(Float32(u2 * fma(u2, Float32(u2 * Float32(Float32(pi) * Float32(Float32(-1.3333333333333333) * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(pi) * Float32(2.0)))) * fma(sqrt(Float32(u1 * Float32(u1 * u1))), Float32(0.25), sqrt(u1)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.150000006

   1. Initial program 56.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 50.3%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      11. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      12. PI-lowering-PI.f32 N/A

          \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

      14. PI-lowering-PI.f32 50.4

          \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right) \]

   7. Simplified 50.4%

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{{u1}^{3}} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{4} \cdot \left(\sqrt{{u1}^{3}} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{\left(\frac{1}{4} \cdot \sqrt{{u1}^{3}}\right) \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. distribute-rgt-out N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{u1} + \frac{1}{4} \cdot \sqrt{{u1}^{3}}\right)} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{u1} + \frac{1}{4} \cdot \sqrt{{u1}^{3}}\right)} \]

   10. Simplified 88.4%

       \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), 2 \cdot \pi\right)\right) \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)} \]

   if 0.150000006 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

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

   5. Simplified 75.5%

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

      1. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

      4. count-2 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. sin-lowering-sin.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      6. distribute-lft-out N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(u2 + u2\right)\right) \]

      9. +-lowering-+.f32 75.5

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

   7. Applied egg-rr 75.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.15000000596046448:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 2\right)\right) \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 2\right)\right) \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (* u2 (fma u2 (* u2 (* PI (* -1.3333333333333333 (* PI PI)))) (* PI 2.0)))
  (fma (sqrt (* u1 (* u1 u1))) 0.25 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * fmaf(u2, (u2 * (((float) M_PI) * (-1.3333333333333333f * (((float) M_PI) * ((float) M_PI))))), (((float) M_PI) * 2.0f))) * fmaf(sqrtf((u1 * (u1 * u1))), 0.25f, sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * fma(u2, Float32(u2 * Float32(Float32(pi) * Float32(Float32(-1.3333333333333333) * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(pi) * Float32(2.0)))) * fma(sqrt(Float32(u1 * Float32(u1 * u1))), Float32(0.25), sqrt(u1)))
end

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 50.1%

   \[\leadsto \sqrt{\color{blue}{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. PI-lowering-PI.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. PI-lowering-PI.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   14. PI-lowering-PI.f32 46.7

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right) \]

7. Simplified 46.7%

   \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\sqrt{{u1}^{3}} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{4} \cdot \left(\sqrt{{u1}^{3}} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{\left(\frac{1}{4} \cdot \sqrt{{u1}^{3}}\right) \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{u1} + \frac{1}{4} \cdot \sqrt{{u1}^{3}}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{u1} + \frac{1}{4} \cdot \sqrt{{u1}^{3}}\right)} \]

10. Simplified 80.4%

    \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), 2 \cdot \pi\right)\right) \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)} \]

11. Final simplification 80.4%

    \[\leadsto \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 2\right)\right) \cdot \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 12: 70.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 2\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma u2 (* u2 (* PI (* -1.3333333333333333 (* PI PI)))) (* PI 2.0))
  (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf(u2, (u2 * (((float) M_PI) * (-1.3333333333333333f * (((float) M_PI) * ((float) M_PI))))), (((float) M_PI) * 2.0f)) * (u2 * sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(u2, Float32(u2 * Float32(Float32(pi) * Float32(Float32(-1.3333333333333333) * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(pi) * Float32(2.0))) * Float32(u2 * sqrt(u1)))
end

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 50.1%

   \[\leadsto \sqrt{\color{blue}{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. cube-mult N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. PI-lowering-PI.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. PI-lowering-PI.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   14. PI-lowering-PI.f32 46.7

       \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right) \]

7. Simplified 46.7%

   \[\leadsto \sqrt{\log \left(\frac{\mathsf{fma}\left(u1, 1 + u1, 1\right)}{1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)}\right) + \mathsf{log1p}\left(u1 \cdot \left(u1 \cdot u1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{u1}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \cdot \sqrt{u1} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

10. Simplified 71.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), 2 \cdot \pi\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

11. Final simplification 71.3%

    \[\leadsto \mathsf{fma}\left(u2, u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right), \pi \cdot 2\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 13: 70.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* PI 2.0))
  (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (((float) M_PI) * 2.0f)) * (u2 * sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(pi) * Float32(2.0))) * Float32(u2 * sqrt(u1)))
end

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 74.1%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{\log \left(1 + u1\right)}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + u1\right)}\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{\log \left(1 + u1\right)}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + u1\right)}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{\log \left(1 + u1\right)}} + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + u1\right)}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{\log \left(1 + u1\right)} + \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right)}}\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\log \left(1 + u1\right)} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\log \left(1 + u1\right)} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{\log \left(1 + u1\right)}} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. accelerator-lowering-log1p.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

7. Simplified 69.4%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot u2\right) \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot u2\right) \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   4. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \left(\color{blue}{\sqrt{u1}} \cdot u2\right) \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)} \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   13. PI-lowering-PI.f32 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   16. PI-lowering-PI.f32 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right) \]

   17. PI-lowering-PI.f32 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right) \]

   18. *-lowering-*.f32 N/A

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right) \]

   19. PI-lowering-PI.f32 71.3

       \[\leadsto \left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right) \]

10. Simplified 71.3%

    \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot u2\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)} \]

11. Final simplification 71.3%

    \[\leadsto \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 14: 70.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right) \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* PI 2.0))
   (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (((float) M_PI) * 2.0f)) * sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(pi) * Float32(2.0))) * sqrt(u1)))
end

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 74.1%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{\log \left(1 + u1\right)}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + u1\right)}\right)\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{\log \left(1 + u1\right)}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + u1\right)}\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{\log \left(1 + u1\right)}} + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(1 + u1\right)}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \sqrt{\log \left(1 + u1\right)} + \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right)}}\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\log \left(1 + u1\right)} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   5. *-lowering-*.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{\log \left(1 + u1\right)} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{\log \left(1 + u1\right)}} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. accelerator-lowering-log1p.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

7. Simplified 69.4%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

8. Taylor expanded in u1 around 0

   \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1}} \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. PI-lowering-PI.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   14. PI-lowering-PI.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   15. PI-lowering-PI.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   17. PI-lowering-PI.f32 71.2

       \[\leadsto u2 \cdot \left(\sqrt{u1} \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right) \]

10. Simplified 71.2%

    \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]

11. Final simplification 71.2%

    \[\leadsto u2 \cdot \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), \pi \cdot 2\right) \cdot \sqrt{u1}\right) \]
12. Add Preprocessing

Alternative 15: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

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

5. Simplified 77.2%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sqrt-lowering-sqrt.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. PI-lowering-PI.f32 65.6

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

8. Simplified 65.6%

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

   1. add-log-exp N/A

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

   2. *-un-lft-identity N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \log \left(e^{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}}\right)\right) \]

   3. exp-prod N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \log \color{blue}{\left({\left(e^{1}\right)}^{\mathsf{PI}\left(\right)}\right)}\right) \]

   4. log-pow N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(e^{1}\right)\right)}\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(e^{1}\right)\right)}\right) \]

   6. PI-lowering-PI.f32 N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \log \left(e^{1}\right)\right)\right) \]

   7. log-lowering-log.f32 N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\log \left(e^{1}\right)}\right)\right) \]

   8. exp-1-e N/A

      \[\leadsto \left(\sqrt{u1} \cdot 2\right) \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \log \color{blue}{\mathsf{E}\left(\right)}\right)\right) \]

   9. E-lowering-E.f32 65.4

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

10. Applied egg-rr 65.4%

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

    1. associate-*l* N/A

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \log \mathsf{E}\left(\right)\right)\right)\right)} \]

    2. *-commutative N/A

       \[\leadsto \sqrt{u1} \cdot \left(2 \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \log \mathsf{E}\left(\right)\right) \cdot u2\right)}\right) \]

    3. log-E N/A

       \[\leadsto \sqrt{u1} \cdot \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{1}\right) \cdot u2\right)\right) \]

    4. *-rgt-identity N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. *-lowering-*.f32 N/A

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

    9. *-lowering-*.f32 N/A

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

    10. sqrt-lowering-sqrt.f32 N/A

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

    11. *-commutative N/A

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

    12. *-lowering-*.f32 N/A

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

    13. PI-lowering-PI.f32 65.7

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

12. Applied egg-rr 65.7%

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

13. Final simplification 65.7%

    \[\leadsto \left(\pi \cdot 2\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]
14. Add Preprocessing

Alternative 16: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

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

5. Simplified 77.2%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sqrt-lowering-sqrt.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. PI-lowering-PI.f32 65.6

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

8. Simplified 65.6%

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

9. Final simplification 65.6%

   \[\leadsto \left(\pi \cdot u2\right) \cdot \left(2 \cdot \sqrt{u1}\right) \]
10. Add Preprocessing

Alternative 17: 7.1% accurate, 231.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

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 = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

1. Initial program 56.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

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

5. Simplified 77.2%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f32 N/A

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

   5. sqrt-lowering-sqrt.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. PI-lowering-PI.f32 65.6

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

8. Simplified 65.6%

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

   1. associate-*r* N/A

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

   2. count-2 N/A

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \mathsf{PI}\left(\right) + u2 \cdot \mathsf{PI}\left(\right)\right)} \]

   3. flip-+ N/A

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\frac{\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) - \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}{u2 \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)}} \]

   4. +-inverses N/A

      \[\leadsto \sqrt{u1} \cdot \frac{\color{blue}{0}}{u2 \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{u1} \cdot \frac{\color{blue}{0 - 0}}{u2 \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)} \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{u1} \cdot \frac{\color{blue}{0 \cdot 0} - 0}{u2 \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)} \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{u1} \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{u2 \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)} \]

   8. +-inverses N/A

      \[\leadsto \sqrt{u1} \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{u1} \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

   10. flip-- N/A

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(0 - 0\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{u1} \cdot \color{blue}{0} \]

   12. mul0-rgt 7.1

       \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 7.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))