Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 91.4%

Time: 10.3s

Alternatives: 8

Speedup: 11.6×

Specification

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

Math

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

FPCore

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ t_1 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - u1 \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left(\left(t\_1 \cdot u2\right) \cdot 2\right) \cdot t\_1\right) \cdot {\mathsf{PI}\left(\right)}^{0.25}\right) \cdot {\mathsf{PI}\left(\right)}^{0.08333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left({t\_0}^{2} \cdot \frac{1}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))) (t_1 (cbrt (PI))))
   (if (<= (- 1.0 u1) 0.9998199939727783)
     (*
      (sqrt (- (log (- 1.0 u1))))
      (cos
       (*
        (* (* (* (* t_1 u2) 2.0) t_1) (pow (PI) 0.25))
        (pow (PI) 0.08333333333333333))))
     (* (sqrt (- (- u1))) (* (pow t_0 2.0) (/ 1.0 t_0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999819994

   1. Initial program 89.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. pow1/3 N/A

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

      10. sqrt-pow1 N/A

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

      11. lower-pow.f32 N/A

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

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{6}}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)\right) \cdot 2\right)\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)}\right) \cdot 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\frac{1}{2}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{\sqrt{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      16. lift-cbrt.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      17. pow1/3 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      18. sqrt-pow1 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      19. lower-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      20. metadata-eval 89.1

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{0.16666666666666666}} \cdot u2\right)\right) \cdot 2\right)\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot u2\right)\right)} \cdot 2\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)} \]

      3. lift-pow.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}\right) \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \]

      6. lift-cbrt.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}\right) \]

      7. pow1/3 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\left(2 \cdot \frac{1}{6}\right)}}\right) \]

      9. pow-sqr N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right) \]

      10. lift-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right) \]

      11. sqr-pow N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)}\right) \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)} \]

      14. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left(\left(\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)} \]

   8. Applied rewrites 89.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left(\left(\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot {\mathsf{PI}\left(\right)}^{0.25}\right) \cdot {\mathsf{PI}\left(\right)}^{0.08333333333333333}\right)} \]

   if 0.999819994 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 40.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower-sin.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lower--.f32 N/A

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f32 N/A

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

   9. Applied rewrites 91.2%

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

Alternative 2: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left({\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot u2\right) \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left({t\_0}^{2} \cdot \frac{1}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))))
   (if (<= (- 1.0 u1) 0.9998199939727783)
     (*
      (sqrt (- (log (- 1.0 u1))))
      (cos
       (*
        (* (pow (PI) 0.8333333333333334) u2)
        (* (pow (PI) 0.16666666666666666) 2.0))))
     (* (sqrt (- (- u1))) (* (pow t_0 2.0) (/ 1.0 t_0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999819994

   1. Initial program 89.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. pow1/3 N/A

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

      10. sqrt-pow1 N/A

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

      11. lower-pow.f32 N/A

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

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{6}}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)\right) \cdot 2\right)\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)}\right) \cdot 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\frac{1}{2}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{\sqrt{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      16. lift-cbrt.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      17. pow1/3 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      18. sqrt-pow1 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      19. lower-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      20. metadata-eval 89.1

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{0.16666666666666666}} \cdot u2\right)\right) \cdot 2\right)\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot u2\right)\right)} \cdot 2\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)} \cdot 2\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot 2\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot 2\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot 2\right)\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot 2\right)\right)} \]

   8. Applied rewrites 89.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{0.8333333333333334} \cdot u2\right) \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot 2\right)\right)} \]

   if 0.999819994 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 40.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower-sin.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lower--.f32 N/A

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f32 N/A

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

   9. Applied rewrites 91.2%

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

Alternative 3: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left({t\_0}^{2} \cdot \frac{1}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))))
   (if (<= (- 1.0 u1) 0.9998199939727783)
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (PI) (+ u2 u2))))
     (* (sqrt (- (- u1))) (* (pow t_0 2.0) (/ 1.0 t_0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999819994

   1. Initial program 89.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. pow1/3 N/A

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

      10. sqrt-pow1 N/A

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

      11. lower-pow.f32 N/A

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

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{6}}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)\right) \cdot 2\right)\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)}\right) \cdot 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\frac{1}{2}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{\sqrt{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      16. lift-cbrt.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      17. pow1/3 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      18. sqrt-pow1 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      19. lower-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      20. metadata-eval 89.1

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{0.16666666666666666}} \cdot u2\right)\right) \cdot 2\right)\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot u2\right)\right)} \cdot 2\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)\right) \cdot 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)\right)\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)}\right)\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot u2\right)}\right)\right) \]

      8. lift-pow.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot u2\right)\right)\right) \]

      9. lift-pow.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right) \cdot u2\right)\right)\right) \]

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. pow1/3 N/A

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

      13. lift-cbrt.f32 N/A

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

      14. associate-*r* N/A

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

   8. Applied rewrites 89.0%

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

   if 0.999819994 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 40.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower-sin.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lower--.f32 N/A

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f32 N/A

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

   9. Applied rewrites 91.2%

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

Alternative 4: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.014919999986886978:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.014919999986886978) (* (sqrt u1) t_1) (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.01492

   1. Initial program 42.1%

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 89.8

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

   7. Applied rewrites 89.8%

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

   if 0.01492 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 89.7%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 78.0%

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

Alternative 5: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.010999999940395355)
     (* (sqrt (- (- u1))) 1.0)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0109999999

   1. Initial program 40.2%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in u2 around 0

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

   8. Applied rewrites 78.3%

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

   if 0.0109999999 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 88.2%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 76.5%

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

Alternative 6: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998199939727783)
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (PI) (+ u2 u2))))
   (* (sqrt u1) (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999819994

   1. Initial program 89.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 89.0

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. sqr-pow N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. pow1/2 N/A

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

      8. lift-cbrt.f32 N/A

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

      9. pow1/3 N/A

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

      10. sqrt-pow1 N/A

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

      11. lower-pow.f32 N/A

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

      12. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{6}}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)\right) \cdot 2\right)\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(\frac{1}{2}\right)} \cdot u2\right)}\right) \cdot 2\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{\frac{1}{2}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{\sqrt{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      16. lift-cbrt.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      17. pow1/3 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\sqrt{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}}}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      18. sqrt-pow1 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      19. lower-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      20. metadata-eval 89.1

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{0.16666666666666666}} \cdot u2\right)\right) \cdot 2\right)\right) \]

   6. Applied rewrites 89.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot u2\right)\right)} \cdot 2\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right) \cdot 2\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)\right) \cdot 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)\right)\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)\right)}\right)\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot u2\right)}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot u2\right)}\right)\right) \]

      8. lift-pow.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot u2\right)\right)\right) \]

      9. lift-pow.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(2 \cdot \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right) \cdot u2\right)\right)\right) \]

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. pow1/3 N/A

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

      13. lift-cbrt.f32 N/A

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

      14. associate-*r* N/A

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

   8. Applied rewrites 89.0%

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

   if 0.999819994 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 40.0%

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

   4. Applied rewrites 61.4%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 91.2

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

   7. Applied rewrites 91.2%

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

Alternative 7: 64.5% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.0

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 66.5%

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

Alternative 8: 4.9% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.0

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

5. Applied rewrites 75.0%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 66.5%

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

9. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot 1 \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. rem-square-sqrt N/A

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

    4. lower-*.f32 N/A

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

    5. lower-sqrt.f32 4.6

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

11. Applied rewrites 4.6%

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

13. Applied rewrites 4.6%

    \[\leadsto \left(-\sqrt{u1}\right) \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2))))