Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.2% → 91.5%

Time: 9.5s

Alternatives: 9

Speedup: 14.4×

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: 57.2% 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: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := {t\_0}^{2}\\ \mathbf{if}\;1 - u1 \leq 0.9998239874839783:\\ \;\;\;\;\cos \left(\sqrt[3]{t\_0} \cdot \left(\sqrt[3]{t\_1} \cdot \left(\left(u2 \cdot t\_1\right) \cdot 2\right)\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right) \cdot {\mathsf{PI}\left(\right)}^{0.7083333333333334}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))) (t_1 (pow t_0 2.0)))
   (if (<= (- 1.0 u1) 0.9998239874839783)
     (*
      (cos (* (cbrt t_0) (* (cbrt t_1) (* (* u2 t_1) 2.0))))
      (sqrt (- (log (- 1.0 u1)))))
     (*
      (sqrt u1)
      (cos
       (*
        (* (* (pow (PI) 0.2916666666666667) u2) 2.0)
        (pow (PI) 0.7083333333333334)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999823987

   1. Initial program 89.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-sqrt.f32 N/A

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

      6. rem-square-sqrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. lift-cbrt.f32 N/A

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

      9. pow3 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f32 N/A

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

      12. associate-*r* N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

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

      15. pow-pow N/A

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

      16. pow3 N/A

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

   6. Applied rewrites 89.5%

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

   if 0.999823987 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 38.3%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-sqrt.f32 3.6

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

   5. Applied rewrites 3.6%

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

   7. Applied rewrites 92.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f32 N/A

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

      12. lift-sqrt.f32 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow1/3 N/A

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

      17. lift-cbrt.f32 N/A

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

      18. lift-pow.f32 N/A

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

   9. Applied rewrites 92.2%

      \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.4166666666666667} \cdot \left({\mathsf{PI}\left(\right)}^{0.5833333333333334} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{7}{12}} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]

       2. lift-*.f32 N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{7}{12}} \cdot \left(2 \cdot u2\right)\right)}\right) \cdot \sqrt{u1} \]

       3. lift-pow.f32 N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{7}{12}}} \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sqrt{u1} \]

       4. sqr-pow N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)} \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sqrt{u1} \]

       5. associate-*l* N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)\right)}\right) \cdot \sqrt{u1} \]

       6. associate-*r* N/A

          \[\leadsto \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]

       7. *-commutative N/A

          \[\leadsto \cos \left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \color{blue}{\left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)}\right) \cdot \sqrt{u1} \]

       8. lower-*.f32 N/A

          \[\leadsto \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right)} \cdot \sqrt{u1} \]

       9. lift-pow.f32 N/A

          \[\leadsto \cos \left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{5}{12}}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       10. pow-prod-up N/A

           \[\leadsto \cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \frac{\frac{7}{12}}{2}\right)}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       11. lower-pow.f32 N/A

           \[\leadsto \cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \frac{\frac{7}{12}}{2}\right)}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       12. metadata-eval N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \color{blue}{\frac{7}{24}}\right)} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       13. metadata-eval N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{17}{24}}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       14. *-commutative N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)}\right) \cdot \sqrt{u1} \]

       15. lift-*.f32 N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \color{blue}{\left(2 \cdot u2\right)}\right)\right) \cdot \sqrt{u1} \]

       16. *-commutative N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \color{blue}{\left(u2 \cdot 2\right)}\right)\right) \cdot \sqrt{u1} \]

       17. associate-*r* N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot u2\right) \cdot 2\right)}\right) \cdot \sqrt{u1} \]

       18. lower-*.f32 N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot u2\right) \cdot 2\right)}\right) \cdot \sqrt{u1} \]

   11. Applied rewrites 92.2%

       \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.7083333333333334} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right)\right)} \cdot \sqrt{u1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998239874839783:\\ \;\;\;\;\cos \left(\sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\left(u2 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot 2\right)\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right) \cdot {\mathsf{PI}\left(\right)}^{0.7083333333333334}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998239874839783:\\ \;\;\;\;\cos \left(\left(\left(u2 \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right) \cdot {\mathsf{PI}\left(\right)}^{0.7083333333333334}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998239874839783)
   (*
    (cos (* (* (* u2 2.0) (cbrt (* (PI) (PI)))) (cbrt (PI))))
    (sqrt (- (log (- 1.0 u1)))))
   (*
    (sqrt u1)
    (cos
     (*
      (* (* (pow (PI) 0.2916666666666667) u2) 2.0)
      (pow (PI) 0.7083333333333334))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999823987

   1. Initial program 89.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f32 N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow1/3 N/A

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

      9. lift-cbrt.f32 N/A

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

      10. sqrt-pow2 N/A

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

      11. lift-sqrt.f32 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-sqrt.f32 N/A

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

      16. sqrt-pow2 N/A

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

      17. lower-pow.f32 N/A

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

      18. metadata-eval 89.5

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

      19. lift-*.f32 N/A

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

      20. *-commutative N/A

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

   6. Applied rewrites 89.5%

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

   7. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cbrt.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-PI.f32 89.5

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

   9. Applied rewrites 89.5%

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

   if 0.999823987 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 38.3%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-sqrt.f32 3.6

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

   5. Applied rewrites 3.6%

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

   7. Applied rewrites 92.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f32 N/A

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

      12. lift-sqrt.f32 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow1/3 N/A

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

      17. lift-cbrt.f32 N/A

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

      18. lift-pow.f32 N/A

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

   9. Applied rewrites 92.2%

      \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.4166666666666667} \cdot \left({\mathsf{PI}\left(\right)}^{0.5833333333333334} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{7}{12}} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]

       2. lift-*.f32 N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{7}{12}} \cdot \left(2 \cdot u2\right)\right)}\right) \cdot \sqrt{u1} \]

       3. lift-pow.f32 N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{7}{12}}} \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sqrt{u1} \]

       4. sqr-pow N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)} \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sqrt{u1} \]

       5. associate-*l* N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)\right)}\right) \cdot \sqrt{u1} \]

       6. associate-*r* N/A

          \[\leadsto \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]

       7. *-commutative N/A

          \[\leadsto \cos \left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \color{blue}{\left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)}\right) \cdot \sqrt{u1} \]

       8. lower-*.f32 N/A

          \[\leadsto \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right)} \cdot \sqrt{u1} \]

       9. lift-pow.f32 N/A

          \[\leadsto \cos \left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{5}{12}}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       10. pow-prod-up N/A

           \[\leadsto \cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \frac{\frac{7}{12}}{2}\right)}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       11. lower-pow.f32 N/A

           \[\leadsto \cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \frac{\frac{7}{12}}{2}\right)}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       12. metadata-eval N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \color{blue}{\frac{7}{24}}\right)} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       13. metadata-eval N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{17}{24}}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       14. *-commutative N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)}\right) \cdot \sqrt{u1} \]

       15. lift-*.f32 N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \color{blue}{\left(2 \cdot u2\right)}\right)\right) \cdot \sqrt{u1} \]

       16. *-commutative N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \color{blue}{\left(u2 \cdot 2\right)}\right)\right) \cdot \sqrt{u1} \]

       17. associate-*r* N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot u2\right) \cdot 2\right)}\right) \cdot \sqrt{u1} \]

       18. lower-*.f32 N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot u2\right) \cdot 2\right)}\right) \cdot \sqrt{u1} \]

   11. Applied rewrites 92.2%

       \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.7083333333333334} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right)\right)} \cdot \sqrt{u1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998239874839783:\\ \;\;\;\;\cos \left(\left(\left(u2 \cdot 2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right) \cdot {\mathsf{PI}\left(\right)}^{0.7083333333333334}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq 0.00017600000137463212:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(t\_1 \cdot \left(u2 \cdot 2\right)\right) \cdot t\_1\right) \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sqrt (PI))))
   (if (<= t_0 0.00017600000137463212)
     (* (cos (* (* 2.0 (PI)) u2)) (sqrt u1))
     (* (cos (* (* t_1 (* u2 2.0)) t_1)) (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.76000001e-4

   1. Initial program 38.3%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-sqrt.f32 3.6

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

   5. Applied rewrites 3.6%

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

   7. Applied rewrites 92.2%

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

   if 1.76000001e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 89.5

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

   4. Applied rewrites 89.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00017600000137463212:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - u1 \leq 0.9998239874839783:\\ \;\;\;\;\cos \left(\left(t\_0 \cdot \left(u2 \cdot 2\right)\right) \cdot t\_0\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right) \cdot {\mathsf{PI}\left(\right)}^{0.7083333333333334}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (PI))))
   (if (<= (- 1.0 u1) 0.9998239874839783)
     (* (cos (* (* t_0 (* u2 2.0)) t_0)) (sqrt (- (log (- 1.0 u1)))))
     (*
      (sqrt u1)
      (cos
       (*
        (* (* (pow (PI) 0.2916666666666667) u2) 2.0)
        (pow (PI) 0.7083333333333334)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999823987

   1. Initial program 89.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 89.5

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

   4. Applied rewrites 89.5%

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

   if 0.999823987 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 38.3%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-sqrt.f32 3.6

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

   5. Applied rewrites 3.6%

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

   7. Applied rewrites 92.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. lift-sqrt.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f32 N/A

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

      12. lift-sqrt.f32 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. pow-prod-up N/A

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

      16. pow1/3 N/A

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

      17. lift-cbrt.f32 N/A

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

      18. lift-pow.f32 N/A

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

   9. Applied rewrites 92.2%

      \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.4166666666666667} \cdot \left({\mathsf{PI}\left(\right)}^{0.5833333333333334} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]
   10. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{7}{12}} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]

       2. lift-*.f32 N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{7}{12}} \cdot \left(2 \cdot u2\right)\right)}\right) \cdot \sqrt{u1} \]

       3. lift-pow.f32 N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{7}{12}}} \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sqrt{u1} \]

       4. sqr-pow N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)} \cdot \left(2 \cdot u2\right)\right)\right) \cdot \sqrt{u1} \]

       5. associate-*l* N/A

          \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)\right)}\right) \cdot \sqrt{u1} \]

       6. associate-*r* N/A

          \[\leadsto \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)\right)} \cdot \sqrt{u1} \]

       7. *-commutative N/A

          \[\leadsto \cos \left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \color{blue}{\left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)}\right) \cdot \sqrt{u1} \]

       8. lower-*.f32 N/A

          \[\leadsto \cos \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\frac{5}{12}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right)} \cdot \sqrt{u1} \]

       9. lift-pow.f32 N/A

          \[\leadsto \cos \left(\left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{5}{12}}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right) \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       10. pow-prod-up N/A

           \[\leadsto \cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \frac{\frac{7}{12}}{2}\right)}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       11. lower-pow.f32 N/A

           \[\leadsto \cos \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \frac{\frac{7}{12}}{2}\right)}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       12. metadata-eval N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\left(\frac{5}{12} + \color{blue}{\frac{7}{24}}\right)} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       13. metadata-eval N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{17}{24}}} \cdot \left(\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)}\right)\right) \cdot \sqrt{u1} \]

       14. *-commutative N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \left(2 \cdot u2\right)\right)}\right) \cdot \sqrt{u1} \]

       15. lift-*.f32 N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \color{blue}{\left(2 \cdot u2\right)}\right)\right) \cdot \sqrt{u1} \]

       16. *-commutative N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot \color{blue}{\left(u2 \cdot 2\right)}\right)\right) \cdot \sqrt{u1} \]

       17. associate-*r* N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot u2\right) \cdot 2\right)}\right) \cdot \sqrt{u1} \]

       18. lower-*.f32 N/A

           \[\leadsto \cos \left({\mathsf{PI}\left(\right)}^{\frac{17}{24}} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{7}{12}}{2}\right)} \cdot u2\right) \cdot 2\right)}\right) \cdot \sqrt{u1} \]

   11. Applied rewrites 92.2%

       \[\leadsto \cos \color{blue}{\left({\mathsf{PI}\left(\right)}^{0.7083333333333334} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right)\right)} \cdot \sqrt{u1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998239874839783:\\ \;\;\;\;\cos \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left({\mathsf{PI}\left(\right)}^{0.2916666666666667} \cdot u2\right) \cdot 2\right) \cdot {\mathsf{PI}\left(\right)}^{0.7083333333333334}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.00017600000137463212:\\ \;\;\;\;t\_1 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= t_0 0.00017600000137463212) (* t_1 (sqrt u1)) (* t_1 (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.76000001e-4

   1. Initial program 38.3%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 N/A

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

      6. lower-sqrt.f32 3.6

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

   5. Applied rewrites 3.6%

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

   7. Applied rewrites 92.2%

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

   if 1.76000001e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00017600000137463212:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.5

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

5. Applied rewrites 3.5%

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

7. Applied rewrites 78.5%

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

8. Final simplification 78.5%

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

Alternative 7: 64.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(u1 \cdot u1\right)}^{0.25} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (pow (* u1 u1) 0.25) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 * u1), 0.25f) * 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 = ((u1 * u1) ** 0.25e0) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(u1 * u1) ^ Float32(0.25)) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((u1 * u1) ^ single(0.25)) * single(1.0);
end

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.5

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

5. Applied rewrites 3.5%

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 69.9%

    \[\leadsto \color{blue}{1} \cdot \sqrt{u1} \]
11. Step-by-step derivation

12. Applied rewrites 69.9%

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

13. Final simplification 69.9%

    \[\leadsto {\left(u1 \cdot u1\right)}^{0.25} \cdot 1 \]
14. Add Preprocessing

Alternative 8: 64.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{-1}{\sqrt{u1}}} \cdot 1 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (-1.0f / (-1.0f / 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 = ((-1.0e0) / ((-1.0e0) / sqrt(u1))) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.5

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

5. Applied rewrites 3.5%

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 69.9%

    \[\leadsto \color{blue}{1} \cdot \sqrt{u1} \]
11. Step-by-step derivation

12. Applied rewrites 69.9%

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

13. Final simplification 69.9%

    \[\leadsto \frac{-1}{\frac{-1}{\sqrt{u1}}} \cdot 1 \]
14. Add Preprocessing

Alternative 9: 64.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{u1} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.5

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

5. Applied rewrites 3.5%

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 69.9%

    \[\leadsto \color{blue}{1} \cdot \sqrt{u1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(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))))