Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.1% → 98.1%

Time: 10.9s

Alternatives: 10

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

Initial Program: 58.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ t_1 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq -0.04500000178813934:\\ \;\;\;\;\sqrt{-t\_1} \cdot \left(\left(\cos t\_0 \cdot \sin t\_0\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)}^{4}\right)}^{0.125} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))) (t_1 (log (- 1.0 u1))))
   (if (<= t_1 -0.04500000178813934)
     (* (sqrt (- t_1)) (* (* (cos t_0) (sin t_0)) 2.0))
     (*
      (pow
       (pow
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)
        4.0)
       0.125)
      (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0450000018

   1. Initial program 97.4%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 97.7

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

   4. Applied rewrites 97.7%

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

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 98.3

         \[\leadsto {\color{blue}{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-pow.f32 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

   9. Applied rewrites 98.4%

      \[\leadsto \color{blue}{{\left({\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)}^{4}\right)}^{0.125}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ t_1 := \log \left(1 - u1\right)\\ t_2 := \left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\\ \mathbf{if}\;t\_1 \leq -0.04500000178813934:\\ \;\;\;\;\sqrt{-t\_1} \cdot \left(\left(\cos t\_0 \cdot \sin t\_0\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_2 \cdot t\_2\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI)))
        (t_1 (log (- 1.0 u1)))
        (t_2
         (*
          (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
          u1)))
   (if (<= t_1 -0.04500000178813934)
     (* (sqrt (- t_1)) (* (* (cos t_0) (sin t_0)) 2.0))
     (* (pow (* t_2 t_2) 0.25) (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0450000018

   1. Initial program 97.4%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 97.7

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

   4. Applied rewrites 97.7%

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

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 98.3

         \[\leadsto {\color{blue}{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.04500000178813934:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\\ t_2 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.04500000178813934:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_1 \cdot t\_1\right)}^{0.25} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1)))
        (t_1
         (*
          (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
          u1))
        (t_2 (sin (* (* 2.0 (PI)) u2))))
   (if (<= t_0 -0.04500000178813934)
     (* (sqrt (- t_0)) t_2)
     (* (pow (* t_1 t_1) 0.25) t_2))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0450000018

   1. Initial program 97.4%

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

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 98.3

         \[\leadsto {\color{blue}{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.04500000178813934:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.04500000178813934:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (* 2.0 (PI)) u2))))
   (if (<= t_0 -0.04500000178813934)
     (* (sqrt (- t_0)) t_1)
     (*
      (sqrt
       (-
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0450000018

   1. Initial program 97.4%

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

   if -0.0450000018 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.07999999821186066:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.07999999821186066)
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (* (* (PI) u2) 2.0))
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0799999982

   1. Initial program 97.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 85.9

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

   7. Applied rewrites 85.9%

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

   if -0.0799999982 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 53.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.03200000151991844:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.03200000151991844)
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (* (* (PI) u2) 2.0))
   (*
    (sqrt (- (* (- (* (- (* -0.3333333333333333 u1) 0.5) u1) 1.0) u1)))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0320000015

   1. Initial program 97.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 84.8

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

   7. Applied rewrites 84.8%

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

   if -0.0320000015 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lower--.f32 N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f32 97.9

          \[\leadsto \sqrt{-\left(\left(\color{blue}{-0.3333333333333333 \cdot u1} - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 97.9%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.01600000075995922:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.01600000075995922)
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (* (* (PI) u2) 2.0))
   (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0160000008

   1. Initial program 96.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 84.8

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

   7. Applied rewrites 84.8%

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

   if -0.0160000008 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 51.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. lower--.f32 N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f32 96.0

          \[\leadsto \sqrt{-\left(\color{blue}{-0.5 \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 96.0%

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

Alternative 8: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.0003000000142492354:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.0003000000142492354)
   (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (* (* (PI) u2) 2.0))
   (* (sqrt u1) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -3.00000014e-4

   1. Initial program 91.0%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 76.4

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

   7. Applied rewrites 76.4%

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

   if -3.00000014e-4 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 42.3%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. metadata-eval 15.1

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

   4. Applied rewrites 14.2%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 90.3

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

   7. Applied rewrites 90.3%

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

Alternative 9: 76.1% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.9%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift--.f32 N/A

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

   9. *-lft-identity N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. metadata-eval 14.4

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

4. Applied rewrites 14.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.4

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

7. Applied rewrites 76.4%

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

Alternative 10: 66.3% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.9%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift--.f32 N/A

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

   9. *-lft-identity N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. metadata-eval 14.4

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

4. Applied rewrites 16.4%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.4

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

7. Applied rewrites 76.4%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 64.5

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

10. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2024347 
(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))))