Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.3%

Time: 4.8s

Alternatives: 11

Speedup: 4.5×

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: 57.5% 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, 0.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (* 2.0 (* (sin (* PI u2)) (sin (fma (- u2) PI (* PI 0.5)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (2.0f * (sinf((((float) M_PI) * u2)) * sinf(fmaf(-u2, ((float) M_PI), (((float) M_PI) * 0.5f)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * sin(fma(Float32(-u2), Float32(pi), Float32(Float32(pi) * Float32(0.5)))))))
end

Derivation

1. Initial program 57.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. flip-+ N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. +-inverses N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. +-inverses N/A

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

   17. flip-+ N/A

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

   18. count-2-rev N/A

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

   19. lift-*.f32 N/A

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

   20. lower-*.f32 25.1

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

5. Applied rewrites 98.3%

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

   1. lift-sin.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. count-2 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. sin-2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-sin.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-cos.f32 98.2

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 98.2

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

7. Applied rewrites 98.2%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-neg.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f32 98.3

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

9. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-u2, \pi, \pi \cdot 0.5\right)\right)}\right)\right) \]
10. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. flip-+ N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. +-inverses N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. +-inverses N/A

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

   17. flip-+ N/A

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

   18. count-2-rev N/A

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

   19. lift-*.f32 N/A

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

   20. lower-*.f32 25.1

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

5. Applied rewrites 98.3%

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

Alternative 3: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0024999999441206455:\\ \;\;\;\;\sqrt{-t\_0} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(u2 + u2\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0024999999441206455)
     (* (sqrt (- t_0)) (sin (* (+ PI PI) u2)))
     (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* (+ u2 u2) PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0024999999441206455f) {
		tmp = sqrtf(-t_0) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(((u2 + u2) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-t_0)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(u2 + u2) * Float32(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.0024999999441206455))
		tmp = sqrt(-t_0) * sin(((single(pi) + single(pi)) * u2));
	else
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * sin(((u2 + u2) * single(pi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00249999994

   1. Initial program 57.5%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.5

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

   3. Applied rewrites 57.5%

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

   if -0.00249999994 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. lift-*.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. count-2-rev N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 25.1

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.0

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

   8. Applied rewrites 88.0%

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

Alternative 4: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00018200000340584666:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(u2 + u2\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00018200000340584666)
   (* (sqrt (- (log1p (- u1)))) (* (+ PI PI) u2))
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* (+ u2 u2) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.00018200000340584666f) {
		tmp = sqrtf(-log1pf(-u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(((u2 + u2) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.00018200000340584666))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(u2 + u2) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 1.82000003e-4

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. lift-*.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. count-2-rev N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 25.1

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-PI.f32 N/A

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

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(1 + 1\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. flip-+ N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 - 1 \cdot 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 - 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{0}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. +-inverses N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{0} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. +-inverses N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{u2 - u2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. flip-+ N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. count-2 N/A

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

      16. lift-PI.f32 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 16.3

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

   10. Applied rewrites 81.1%

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

   if 1.82000003e-4 < u2

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. lift-*.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. count-2-rev N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 25.1

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.0

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

   8. Applied rewrites 88.0%

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

Alternative 5: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0019000000320374966:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(u2 + u2\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0019000000320374966)
   (* (sqrt (- (log1p (- u1)))) (* (+ PI PI) u2))
   (* (sqrt u1) (sin (* (+ u2 u2) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0019000000320374966f) {
		tmp = sqrtf(-log1pf(-u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
	} else {
		tmp = sqrtf(u1) * sinf(((u2 + u2) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0019000000320374966))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(Float32(pi) + Float32(pi)) * u2));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(u2 + u2) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00190000003

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. lift-*.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. count-2-rev N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 25.1

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.1

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

   8. Applied rewrites 81.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-PI.f32 N/A

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

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(1 + 1\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. flip-+ N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 - 1 \cdot 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 - 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{0}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. +-inverses N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{0} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. +-inverses N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{u2 - u2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. flip-+ N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. count-2 N/A

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

      16. lift-PI.f32 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 16.3

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

   10. Applied rewrites 81.1%

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

   if 0.00190000003 < u2

   1. Initial program 57.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. lift-*.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. flip-+ N/A

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

      18. count-2-rev N/A

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

      19. lift-*.f32 N/A

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

      20. lower-*.f32 25.1

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.6%

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

Alternative 6: 81.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. flip-+ N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. +-inverses N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. metadata-eval N/A

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

   16. +-inverses N/A

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

   17. flip-+ N/A

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

   18. count-2-rev N/A

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

   19. lift-*.f32 N/A

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

   20. lower-*.f32 25.1

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.1

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

8. Applied rewrites 81.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-PI.f32 N/A

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

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(1 + 1\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. flip-+ N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 \cdot 1 - 1 \cdot 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 - 1 \cdot 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{1 - 1}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{0}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. +-inverses N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{1 - 1} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{0} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   13. +-inverses N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\frac{u2 \cdot u2 - u2 \cdot u2}{u2 - u2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. flip-+ N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   15. count-2 N/A

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

   16. lift-PI.f32 N/A

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

   17. associate-*r* N/A

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. lower-*.f32 16.3

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

10. Applied rewrites 81.1%

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

Alternative 7: 80.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 + u2\right)\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0020000000949949026)
   (* (sqrt (* u1 (+ 1.0 (* u1 0.5)))) (* 2.0 (* u2 PI)))
   (* (* (sqrt (- (log (- 1.0 u1)))) (+ u2 u2)) PI)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0020000000949949026f) {
		tmp = sqrtf((u1 * (1.0f + (u1 * 0.5f)))) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = (sqrtf(-logf((1.0f - u1))) * (u2 + u2)) * ((float) M_PI);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 + u2)) * Float32(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.0020000000949949026))
		tmp = sqrt((u1 * (single(1.0) + (u1 * single(0.5))))) * (single(2.0) * (u2 * single(pi)));
	else
		tmp = (sqrt(-log((single(1.0) - u1))) * (u2 + u2)) * single(pi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00200000009

   1. Initial program 57.5%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.4

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.4%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 77.7

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \frac{1}{2}\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

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

   if 0.00200000009 < u1

   1. Initial program 57.5%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.5%

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

Alternative 8: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* u1 (+ 1.0 (* u1 0.5)))) (* 2.0 (* u2 PI))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * 0.5f)))) * (2.0f * (u2 * ((float) M_PI)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * single(0.5))))) * (single(2.0) * (u2 * single(pi)));
end

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 93.4

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Applied rewrites 93.4%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 77.7

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]

7. Applied rewrites 77.7%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \frac{1}{2}\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 73.8%

    \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
11. Add Preprocessing

Alternative 9: 66.0% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 2.0 (* u2 (* u1 (* PI (sqrt (/ 1.0 u1)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.0

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

7. Applied rewrites 66.0%

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

8. Taylor expanded in u1 around inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right)\right) \]

   4. lower-sqrt.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right)\right) \]

   5. lower-/.f32 66.0

      \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right)\right) \]

10. Applied rewrites 66.0%

    \[\leadsto 2 \cdot \left(u2 \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right)\right) \]
11. Add Preprocessing

Alternative 10: 66.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.0

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

7. Applied rewrites 66.0%

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

Alternative 11: 16.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.0

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

7. Applied rewrites 66.0%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-rgt-out N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

      \[\leadsto \left(\pi \cdot \sqrt{u1}\right) \cdot \frac{1 \cdot 1 - 1}{u2 - u2} \]

   10. metadata-eval N/A

       \[\leadsto \left(\pi \cdot \sqrt{u1}\right) \cdot \frac{1 \cdot 1 - 1 \cdot 1}{u2 - u2} \]

   11. +-inverses N/A

       \[\leadsto \left(\pi \cdot \sqrt{u1}\right) \cdot \frac{1 \cdot 1 - 1 \cdot 1}{0} \]

   12. metadata-eval N/A

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

   13. flip-+ N/A

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

   14. metadata-eval N/A

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

9. Applied rewrites 16.0%

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

Reproduce

herbie shell --seed 2025142 
(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))))