Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 96.8%

Time: 12.0s

Alternatives: 11

Speedup: 7.7×

Specification

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

Math

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

FPCore

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ t_1 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_2 := {t\_1}^{2}\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\sin \left(\sqrt[3]{t\_1} \cdot \left(\sqrt[3]{t\_2} \cdot \left(\left(u2 \cdot t\_2\right) \cdot 2\right)\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t\_0, \sqrt{u1} \cdot \left(\left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5\right)\right), u1, 0.25 \cdot t\_0\right) + \sqrt{u1}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))) (t_1 (cbrt (PI))) (t_2 (pow t_1 2.0)))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (*
      (sin (* (cbrt t_1) (* (cbrt t_2) (* (* u2 t_2) 2.0))))
      (sqrt (- (log (- 1.0 u1)))))
     (*
      (sin (* (* 2.0 (PI)) u2))
      (sqrt
       (pow
        (+
         (*
          (* u1 u1)
          (fma
           (fma
            0.16666666666666666
            t_0
            (* (sqrt u1) (* (+ 0.25 (/ -0.0625 u1)) 0.5)))
           u1
           (* 0.25 t_0)))
         (sqrt u1))
        2.0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. pow2 N/A

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

      14. lower-pow.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lower-cbrt.f32 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f32 N/A

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

      2. rem-cbrt-cube N/A

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

      3. cube-unmult N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f32 N/A

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

      6. cbrt-prod N/A

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

      7. pow1/3 N/A

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

      8. pow1/3 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f32 N/A

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

   6. Applied rewrites 93.6%

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

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 88.2

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

   5. Applied rewrites 88.2%

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

      1. +-lft-identity N/A

         \[\leadsto \sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip3-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-lft-identity N/A

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

      4. lift-pow.f32 N/A

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

      5. sqr-pow N/A

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

      6. lift-+.f32 N/A

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

      7. +-lft-identity N/A

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

      8. lift--.f32 N/A

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

   9. Applied rewrites 88.1%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{{\left(-\left(-u1\right)\right)}^{1.5}}{-\left(-u1\right)}\right)}^{2}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sqrt{\frac{1}{u1}}, \left(0.5 \cdot \left(\frac{-0.0625}{u1} + 0.25\right)\right) \cdot \sqrt{u1}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right) \cdot \left(u1 \cdot u1\right) - \left(-\sqrt{u1}\right)\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\sin \left(\sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\left(u2 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot 2\right)\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sqrt{\frac{1}{u1}}, \sqrt{u1} \cdot \left(\left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5\right)\right), u1, 0.25 \cdot \sqrt{\frac{1}{u1}}\right) + \sqrt{u1}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\sin \left(\left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t\_0, \sqrt{u1} \cdot \left(\left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5\right)\right), u1, 0.25 \cdot t\_0\right) + \sqrt{u1}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (*
      (sin (* (* (cbrt (* (PI) (PI))) (* u2 2.0)) (cbrt (PI))))
      (sqrt (- (log (- 1.0 u1)))))
     (*
      (sin (* (* 2.0 (PI)) u2))
      (sqrt
       (pow
        (+
         (*
          (* u1 u1)
          (fma
           (fma
            0.16666666666666666
            t_0
            (* (sqrt u1) (* (+ 0.25 (/ -0.0625 u1)) 0.5)))
           u1
           (* 0.25 t_0)))
         (sqrt u1))
        2.0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. pow2 N/A

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

      14. lower-pow.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lower-cbrt.f32 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cbrt.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-PI.f32 93.6

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

   7. Applied rewrites 93.6%

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

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 88.2

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

   5. Applied rewrites 88.2%

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

      1. +-lft-identity N/A

         \[\leadsto \sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip3-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-lft-identity N/A

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

      4. lift-pow.f32 N/A

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

      5. sqr-pow N/A

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

      6. lift-+.f32 N/A

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

      7. +-lft-identity N/A

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

      8. lift--.f32 N/A

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

   9. Applied rewrites 88.1%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{{\left(-\left(-u1\right)\right)}^{1.5}}{-\left(-u1\right)}\right)}^{2}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sqrt{\frac{1}{u1}}, \left(0.5 \cdot \left(\frac{-0.0625}{u1} + 0.25\right)\right) \cdot \sqrt{u1}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right) \cdot \left(u1 \cdot u1\right) - \left(-\sqrt{u1}\right)\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\sin \left(\left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sqrt{\frac{1}{u1}}, \sqrt{u1} \cdot \left(\left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5\right)\right), u1, 0.25 \cdot \sqrt{\frac{1}{u1}}\right) + \sqrt{u1}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{{\left(-\left(-u1\right)\right)}^{3}}{u1 \cdot u1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 (PI)) u2))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.00019999999494757503)
     (* (sqrt (/ (pow (- (- u1)) 3.0) (* u1 u1))) t_0)
     (* t_0 (sqrt t_1)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.99999995e-4

   1. Initial program 36.7%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.0

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

   5. Applied rewrites 92.0%

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

      1. +-lft-identity N/A

         \[\leadsto \sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip3-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 26.6%

      \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\color{blue}{\frac{{\left(-\left(-u1\right)\right)}^{8} - 0}{{\left(-u1\right)}^{6}}} - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{{u1}^{2}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. lower-*.f32 92.0

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

   12. Applied rewrites 92.0%

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

   if 1.99999995e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{{\left(-\left(-u1\right)\right)}^{3}}{u1 \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := \sqrt{\frac{1}{u1}}\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t\_1, \sqrt{u1} \cdot \left(\left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5\right)\right), u1, 0.25 \cdot t\_1\right) + \sqrt{u1}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 (PI)) u2))) (t_1 (sqrt (/ 1.0 u1))))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (*
      t_0
      (sqrt
       (pow
        (+
         (*
          (* u1 u1)
          (fma
           (fma
            0.16666666666666666
            t_1
            (* (sqrt u1) (* (+ 0.25 (/ -0.0625 u1)) 0.5)))
           u1
           (* 0.25 t_1)))
         (sqrt u1))
        2.0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

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

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 88.2

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

   5. Applied rewrites 88.2%

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

      1. +-lft-identity N/A

         \[\leadsto \sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip3-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-lft-identity N/A

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

      4. lift-pow.f32 N/A

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

      5. sqr-pow N/A

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

      6. lift-+.f32 N/A

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

      7. +-lft-identity N/A

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

      8. lift--.f32 N/A

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

   9. Applied rewrites 88.1%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{{\left(-\left(-u1\right)\right)}^{1.5}}{-\left(-u1\right)}\right)}^{2}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sqrt{\frac{1}{u1}}, \left(0.5 \cdot \left(\frac{-0.0625}{u1} + 0.25\right)\right) \cdot \sqrt{u1}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right) \cdot \left(u1 \cdot u1\right) - \left(-\sqrt{u1}\right)\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, \sqrt{\frac{1}{u1}}, \sqrt{u1} \cdot \left(\left(0.25 + \frac{-0.0625}{u1}\right) \cdot 0.5\right)\right), u1, 0.25 \cdot \sqrt{\frac{1}{u1}}\right) + \sqrt{u1}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sin (* (* 2.0 (PI)) u2))))
   (if (<= t_0 0.00019999999494757503)
     (* (sqrt (/ 1.0 (/ 1.0 u1))) t_1)
     (* t_1 (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.99999995e-4

   1. Initial program 36.7%

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

   4. Applied rewrites 35.7%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f32 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f32 N/A

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

      8. lower-/.f32 39.0

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

      9. lift--.f32 N/A

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

      10. lift-log1p.f32 N/A

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

      11. lift-log1p.f32 N/A

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

      12. diff-log N/A

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

      13. clear-num N/A

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

   6. Applied rewrites 39.1%

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

   7. Taylor expanded in u1 around 0

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

      1. lower-/.f32 92.0

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

   9. Applied rewrites 92.0%

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

   if 1.99999995e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) u2)))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt (/ 1.0 (/ 1.0 u1))) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 80.7

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

   5. Applied rewrites 80.7%

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

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

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

   4. Applied rewrites 32.8%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f32 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f32 N/A

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

      8. lower-/.f32 35.1

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

      9. lift--.f32 N/A

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

      10. lift-log1p.f32 N/A

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

      11. lift-log1p.f32 N/A

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

      12. diff-log N/A

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

      13. clear-num N/A

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

   6. Applied rewrites 37.9%

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

   7. Taylor expanded in u1 around 0

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

      1. lower-/.f32 88.3

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

   9. Applied rewrites 88.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) u2)))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 80.7

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

   5. Applied rewrites 80.7%

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

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

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

   4. Applied rewrites 15.8%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

4. Applied rewrites 13.6%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.1

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

7. Applied rewrites 76.1%

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

Alternative 9: 65.9% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 76.1

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

5. Applied rewrites 76.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.9

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

8. Applied rewrites 63.9%

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

10. Applied rewrites 63.9%

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

11. Final simplification 63.9%

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

Alternative 10: 65.9% accurate, 7.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 76.1

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

5. Applied rewrites 76.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.9

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

8. Applied rewrites 63.9%

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

9. Final simplification 63.9%

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

Alternative 11: 4.7% accurate, 8.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 4.0

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 4.8

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

8. Applied rewrites 4.8%

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

9. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))