Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.2% → 91.2%

Time: 8.5s

Alternatives: 6

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: 57.2% 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: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{u1}}{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998499751091003)
   (* (sqrt (- (log (- 1.0 u1)))) (sin (* (PI) (+ u2 u2))))
   (* (/ 1.0 (/ (sqrt u1) u1)) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999849975

   1. Initial program 89.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f32 N/A

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

      7. pow-plus N/A

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

      8. lift-cbrt.f32 N/A

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

      9. metadata-eval N/A

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

      10. rem-cube-cbrt N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2 N/A

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

      15. lift-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f32 N/A

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

      19. lower-+.f32 89.8

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

   6. Applied rewrites 89.8%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 37.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 4.2

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

   5. Applied rewrites 4.2%

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

   7. Applied rewrites 92.9%

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

Alternative 2: 44.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998999834060669:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\cos t\_0 \cdot \sin t\_0\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right), u1 \cdot u1, -1\right) \cdot \left(u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))))
   (if (<= (- 1.0 u1) 0.9998999834060669)
     (* (sqrt (- (log (- 1.0 u1)))) (* (* (cos t_0) (sin t_0)) 2.0))
     (*
      (sqrt
       (-
        (-
         (*
          (fma (fma -0.3333333333333333 (* u1 u1) -0.5) (* u1 u1) -1.0)
          (* u1 u1))
         (log1p u1))))
      (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999899983

   1. Initial program 88.9%

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

      1. lift-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 88.9

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

   4. Applied rewrites 88.9%

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

   if 0.999899983 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.4%

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right) \cdot {u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right) \cdot {u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. sub-neg N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) \cdot {u1}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\left(\left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) \cdot {u1}^{2} + \color{blue}{-1}\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}, {u1}^{2}, -1\right)} \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot {u1}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\frac{-1}{3} \cdot {u1}^{2} + \color{blue}{\frac{-1}{2}}, {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {u1}^{2}, \frac{-1}{2}\right)}, {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{u1 \cdot u1}, \frac{-1}{2}\right), {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{u1 \cdot u1}, \frac{-1}{2}\right), {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, u1 \cdot u1, \frac{-1}{2}\right), \color{blue}{u1 \cdot u1}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, u1 \cdot u1, \frac{-1}{2}\right), \color{blue}{u1 \cdot u1}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, u1 \cdot u1, \frac{-1}{2}\right), u1 \cdot u1, -1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-*.f32 24.9

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right), u1 \cdot u1, -1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 28.7%

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right), u1 \cdot u1, -1\right) \cdot \left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998999834060669:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right), u1 \cdot u1, -1\right) \cdot \left(u1 \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998999834060669)
   (* (sqrt (- (log (- 1.0 u1)))) (sin (* (PI) (+ u2 u2))))
   (*
    (sqrt
     (-
      (-
       (*
        (fma (fma -0.3333333333333333 (* u1 u1) -0.5) (* u1 u1) -1.0)
        (* u1 u1))
       (log1p u1))))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999899983

   1. Initial program 88.9%

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

      1. lift-*.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 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f32 N/A

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

      7. pow-plus N/A

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

      8. lift-cbrt.f32 N/A

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

      9. metadata-eval N/A

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

      10. rem-cube-cbrt N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2 N/A

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

      15. lift-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. distribute-rgt-out N/A

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

      18. lower-*.f32 N/A

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

      19. lower-+.f32 88.9

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

   6. Applied rewrites 88.9%

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

   if 0.999899983 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.4%

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right) \cdot {u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right) \cdot {u1}^{2}} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. sub-neg N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) \cdot {u1}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\left(\left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) \cdot {u1}^{2} + \color{blue}{-1}\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}, {u1}^{2}, -1\right)} \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. sub-neg N/A

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot {u1}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\frac{-1}{3} \cdot {u1}^{2} + \color{blue}{\frac{-1}{2}}, {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {u1}^{2}, \frac{-1}{2}\right)}, {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{u1 \cdot u1}, \frac{-1}{2}\right), {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{u1 \cdot u1}, \frac{-1}{2}\right), {u1}^{2}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, u1 \cdot u1, \frac{-1}{2}\right), \color{blue}{u1 \cdot u1}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, u1 \cdot u1, \frac{-1}{2}\right), \color{blue}{u1 \cdot u1}, -1\right) \cdot {u1}^{2} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. unpow2 N/A

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, u1 \cdot u1, \frac{-1}{2}\right), u1 \cdot u1, -1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-*.f32 28.5

          \[\leadsto \sqrt{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right), u1 \cdot u1, -1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 25.6%

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right), u1 \cdot u1, -1\right) \cdot \left(u1 \cdot u1\right)} - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 76.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.7%

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

7. Applied rewrites 76.5%

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

Alternative 5: 76.7% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.7%

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

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

7. Applied rewrites 76.5%

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

Alternative 6: 66.2% 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 58.7%

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

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

7. Applied rewrites 76.5%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 68.0

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

10. Applied rewrites 68.0%

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

Reproduce

herbie shell --seed 2024322 
(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))))