Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.0% → 91.2%

Time: 10.1s

Alternatives: 9

Speedup: 11.6×

Specification

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

Math

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

FPCore

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

Initial Program: 57.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq 0.00015500000154133886:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \cos \left(\left(\left(2 \cdot u2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \cos \left({t\_1}^{2} \cdot \left(\left(t\_1 \cdot u2\right) \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (cbrt (PI))))
   (if (<= t_0 0.00015500000154133886)
     (* (sqrt (- (- u1))) (cos (* (* (* 2.0 u2) (cbrt (* (PI) (PI)))) t_1)))
     (* (sqrt t_0) (cos (* (pow t_1 2.0) (* (* t_1 u2) 2.0)))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.55000002e-4

   1. Initial program 34.1%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cbrt.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cbrt.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-PI.f32 93.5

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

   10. Applied rewrites 93.5%

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

   if 1.55000002e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 90.8

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

   4. Applied rewrites 90.8%

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

Alternative 2: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.00015500000154133886:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \cos \left(\left(\left(2 \cdot u2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\left(t\_1 + 1\right) - \left(1 - t\_1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (cos (* u2 (* (PI) 2.0)))))
   (if (<= t_0 0.00015500000154133886)
     (*
      (sqrt (- (- u1)))
      (cos (* (* (* 2.0 u2) (cbrt (* (PI) (PI)))) (cbrt (PI)))))
     (* (sqrt t_0) (/ (- (+ t_1 1.0) (- 1.0 t_1)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.55000002e-4

   1. Initial program 34.1%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cbrt.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cbrt.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-PI.f32 93.5

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

   10. Applied rewrites 93.5%

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

   if 1.55000002e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.7%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. cos-mult N/A

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

      7. sin-mult N/A

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

      8. sub-div N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 90.7%

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

Alternative 3: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.012600000016391277:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.012600000016391277)
     (* (pow (* u1 u1) 0.25) t_1)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0126

   1. Initial program 36.6%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval 91.5

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

         \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{\frac{1}{4}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 91.5

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

   10. Applied rewrites 91.5%

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

   if 0.0126 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.2%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 80.1%

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

Alternative 4: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.00015500000154133886:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \cos \left(\left(\left(2 \cdot u2\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.00015500000154133886)
     (*
      (sqrt (- (- u1)))
      (cos (* (* (* 2.0 u2) (cbrt (* (PI) (PI)))) (cbrt (PI)))))
     (* (sqrt t_0) (cos (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.55000002e-4

   1. Initial program 34.1%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cbrt.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cbrt.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-PI.f32 93.5

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

   10. Applied rewrites 93.5%

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

   if 1.55000002e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.7%

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

Alternative 5: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00015500000154133886:\\ \;\;\;\;\sqrt{\frac{{\left(-\left(-u1\right)\right)}^{3}}{\left(-u1\right) \cdot \left(-u1\right)}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.00015500000154133886)
     (* (sqrt (/ (pow (- (- u1)) 3.0) (* (- u1) (- u1)))) t_0)
     (* (sqrt t_1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.55000002e-4

   1. Initial program 34.1%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 93.5

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

   5. Applied rewrites 93.5%

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

      1. +-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. flip3-+ N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{{\color{blue}{0}}^{3} + {\left(-\left(-u1\right)\right)}^{3}}{\log 1 \cdot \log 1 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - \log 1 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0} + {\left(-\left(-u1\right)\right)}^{3}}{\log 1 \cdot \log 1 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - \log 1 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

      9. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0} + {\left(-\left(-u1\right)\right)}^{3}}{\log 1 \cdot \log 1 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - \log 1 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \sqrt{\frac{0 + \color{blue}{{\left(-\left(-u1\right)\right)}^{3}}}{\log 1 \cdot \log 1 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - \log 1 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 93.5%

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

   if 1.55000002e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00015500000154133886:\\ \;\;\;\;\sqrt{\frac{{\left(-\left(-u1\right)\right)}^{3}}{\left(-u1\right) \cdot \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.012600000016391277:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.012600000016391277) (* (sqrt u1) t_1) (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0126

   1. Initial program 36.6%

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 91.5

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

   7. Applied rewrites 91.5%

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

   if 0.0126 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.2%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 80.1%

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

Alternative 7: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.012600000016391277:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.012600000016391277)
     (* (sqrt (- (- u1))) 1.0)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0126

   1. Initial program 36.6%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 74.2

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

   8. Applied rewrites 73.9%

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

   9. Taylor expanded in u2 around 0

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

   11. Applied rewrites 74.2%

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

   if 0.0126 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.2%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 80.1%

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

Alternative 8: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.00015500000154133886:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= t_0 0.00015500000154133886)
     (* (pow (* u1 u1) 0.25) t_1)
     (* (sqrt t_0) t_1))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.55000002e-4

   1. Initial program 34.1%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

         \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{\frac{1}{4}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 93.5

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

   10. Applied rewrites 93.5%

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

   if 1.55000002e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.7%

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

Alternative 9: 65.2% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.6

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

5. Applied rewrites 75.6%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 64.4

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

8. Applied rewrites 64.2%

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

9. Taylor expanded in u2 around 0

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

11. Applied rewrites 64.4%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2))))