Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.0% → 90.9%

Time: 10.3s

Alternatives: 9

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

Initial Program: 58.0% 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: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00015999999595806003:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sin \left({t\_0}^{2} \cdot \left(\left(t\_0 \cdot u2\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.00015999999595806003)
     (/
      (- (sin (* (* u2 (PI)) 2.0)))
      (/
       (- (* (* (- -0.041666666666666664) (sqrt u1)) (* u1 u1)) (sqrt u1))
       u1))
     (* (sqrt t_1) (sin (* (pow t_0 2.0) (* (* t_0 u2) 2.0)))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.59999996e-4

   1. Initial program 37.3%

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

   4. Applied rewrites 17.6%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 38.9%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]
   8. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]

   9. Applied rewrites 91.8%

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}} \]

   10. Taylor expanded in u1 around -inf

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\frac{-1}{24} \cdot \left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]
   11. Step-by-step derivation

   12. Applied rewrites 91.8%

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(-0.041666666666666664 \cdot \left(-\sqrt{u1}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

   if 1.59999996e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.8%

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

      1. lift-*.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. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00015999999595806003:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00015999999595806003:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \sin \left(\left(\left(u2 \cdot 2\right) \cdot {t\_0}^{2}\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.00015999999595806003)
     (/
      (- (sin (* (* u2 (PI)) 2.0)))
      (/
       (- (* (* (- -0.041666666666666664) (sqrt u1)) (* u1 u1)) (sqrt u1))
       u1))
     (* (sqrt t_1) (sin (* (* (* u2 2.0) (pow t_0 2.0)) t_0))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.59999996e-4

   1. Initial program 37.3%

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

   4. Applied rewrites 18.9%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 37.0%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]
   8. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]

   9. Applied rewrites 91.8%

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}} \]

   10. Taylor expanded in u1 around -inf

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\frac{-1}{24} \cdot \left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]
   11. Step-by-step derivation

   12. Applied rewrites 91.8%

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(-0.041666666666666664 \cdot \left(-\sqrt{u1}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

   if 1.59999996e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.8%

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

      1. lift-*.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 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00015999999595806003:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\left(u2 \cdot 2\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.00015999999595806003:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.00015999999595806003)
     (/
      (- (sin (* (* u2 (PI)) 2.0)))
      (/
       (- (* (* (- -0.041666666666666664) (sqrt u1)) (* u1 u1)) (sqrt u1))
       u1))
     (* (sqrt t_0) (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.59999996e-4

   1. Initial program 37.3%

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

   4. Applied rewrites 17.6%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 38.3%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]
   8. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]

   9. Applied rewrites 91.8%

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}} \]

   10. Taylor expanded in u1 around -inf

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\frac{-1}{24} \cdot \left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]
   11. Step-by-step derivation

   12. Applied rewrites 91.8%

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(-0.041666666666666664 \cdot \left(-\sqrt{u1}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

   if 1.59999996e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00015999999595806003:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.000699999975040555:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.000699999975040555)
     (/
      (- (sin (* (* u2 (PI)) 2.0)))
      (/
       (- (* (* (- -0.041666666666666664) (sqrt u1)) (* u1 u1)) (sqrt u1))
       u1))
     (* (sqrt t_0) (* (* (PI) u2) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 6.99999975e-4

   1. Initial program 40.7%

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

   4. Applied rewrites 16.7%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 37.4%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]
   8. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]

   9. Applied rewrites 89.7%

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}} \]

   10. Taylor expanded in u1 around -inf

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\frac{-1}{24} \cdot \left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]
   11. Step-by-step derivation

   12. Applied rewrites 89.7%

       \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(-0.041666666666666664 \cdot \left(-\sqrt{u1}\right)\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

   if 6.99999975e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 93.3%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right) \cdot u2\right)} \cdot 2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right) \cdot u2\right)} \cdot 2\right) \]

   7. Applied rewrites 14.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{5} \cdot 0.13333333333333333\right) \cdot u2, u2, {\mathsf{PI}\left(\right)}^{3} \cdot -0.6666666666666666\right), u2 \cdot u2, \mathsf{PI}\left(\right)\right) \cdot u2\right)} \cdot 2\right) \]

   8. Taylor expanded in u2 around 0

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

   10. Applied rewrites 81.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.000699999975040555:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\frac{\left(\left(--0.041666666666666664\right) \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9993000030517578:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{-\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9993000030517578)
   (* (sqrt (- (log (- 1.0 u1)))) (* (* (PI) u2) 2.0))
   (/ (- (sin (* (* u2 (PI)) 2.0))) (- (sqrt (/ 1.0 u1))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.9993

   1. Initial program 93.3%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right) \cdot u2\right)} \cdot 2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right) \cdot u2\right)} \cdot 2\right) \]

   7. Applied rewrites 14.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{5} \cdot 0.13333333333333333\right) \cdot u2, u2, {\mathsf{PI}\left(\right)}^{3} \cdot -0.6666666666666666\right), u2 \cdot u2, \mathsf{PI}\left(\right)\right) \cdot u2\right)} \cdot 2\right) \]

   8. Taylor expanded in u2 around 0

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

   10. Applied rewrites 81.5%

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

   if 0.9993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 40.7%

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

   4. Applied rewrites 15.8%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 37.2%

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

   7. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-/.f32 89.6

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

   9. Applied rewrites 89.6%

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

Alternative 6: 85.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9993000030517578:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9993000030517578)
   (* (sqrt (- (log (- 1.0 u1)))) (* (* (PI) u2) 2.0))
   (* (sin (* (PI) (+ u2 u2))) (sqrt u1))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.9993

   1. Initial program 93.3%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right)\right)} \cdot 2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right) \cdot u2\right)} \cdot 2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-1}{2} \cdot {\mathsf{PI}\left(\right)}^{3} + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{3} + {u2}^{2} \cdot \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{5} + \left(\frac{1}{24} \cdot {\mathsf{PI}\left(\right)}^{5} + \frac{1}{12} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)\right) \cdot u2\right)} \cdot 2\right) \]

   7. Applied rewrites 14.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{5} \cdot 0.13333333333333333\right) \cdot u2, u2, {\mathsf{PI}\left(\right)}^{3} \cdot -0.6666666666666666\right), u2 \cdot u2, \mathsf{PI}\left(\right)\right) \cdot u2\right)} \cdot 2\right) \]

   8. Taylor expanded in u2 around 0

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

   10. Applied rewrites 81.5%

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

   if 0.9993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 40.7%

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

   4. Applied rewrites 16.6%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. lower-sqrt.f32 89.5

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

   7. Applied rewrites 89.5%

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

   9. Applied rewrites 89.5%

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

Alternative 7: 76.5% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.0%

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

4. Applied rewrites 15.7%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-sqrt.f32 75.7

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

7. Applied rewrites 75.7%

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

9. Applied rewrites 75.7%

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

Alternative 8: 50.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/
  (* -2.0 (* (PI) u2))
  (/
   (-
    (*
     (fma -0.25 (sqrt (/ 1.0 u1)) (* 0.041666666666666664 (sqrt u1)))
     (* u1 u1))
    (sqrt u1))
   u1)))

Derivation

1. Initial program 58.0%

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

4. Applied rewrites 13.7%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

6. Applied rewrites 35.0%

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

7. Taylor expanded in u1 around 0

   \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{-1 \cdot \sqrt{u1} + {u1}^{2} \cdot \left(\frac{-1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{24} \cdot \sqrt{u1}\right)}{u1}}} \]

9. Applied rewrites 75.9%

   \[\leadsto \frac{-\sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}{\color{blue}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}}} \]

10. Taylor expanded in u2 around 0

    \[\leadsto \frac{\color{blue}{-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}}{\frac{\mathsf{fma}\left(\frac{-1}{4}, \sqrt{\frac{1}{u1}}, \frac{1}{24} \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)}}{\frac{\mathsf{fma}\left(\frac{-1}{4}, \sqrt{\frac{1}{u1}}, \frac{1}{24} \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}}{\frac{\mathsf{fma}\left(\frac{-1}{4}, \sqrt{\frac{1}{u1}}, \frac{1}{24} \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

    3. lower-*.f32 N/A

       \[\leadsto \frac{-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}}{\frac{\mathsf{fma}\left(\frac{-1}{4}, \sqrt{\frac{1}{u1}}, \frac{1}{24} \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

    4. lower-PI.f32 67.0

       \[\leadsto \frac{-2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u2\right)}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]

12. Applied rewrites 67.0%

    \[\leadsto \frac{\color{blue}{-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}}{\frac{\mathsf{fma}\left(-0.25, \sqrt{\frac{1}{u1}}, 0.041666666666666664 \cdot \sqrt{u1}\right) \cdot \left(u1 \cdot u1\right) - \sqrt{u1}}{u1}} \]
13. Add Preprocessing

Alternative 9: 65.9% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.0%

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

4. Applied rewrites 13.6%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-sqrt.f32 75.7

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

7. Applied rewrites 75.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 66.9%

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

Reproduce

herbie shell --seed 2024307 
(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))))