Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 91.2%

Time: 8.1s

Alternatives: 6

Speedup: 14.4×

Specification

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

Math

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

FPCore

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

Initial Program: 58.0% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - u1 \leq 0.9998940229415894:\\ \;\;\;\;\cos \left(\left(t\_0 \cdot \left(2 \cdot u2\right)\right) \cdot t\_0\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (PI))))
   (if (<= (- 1.0 u1) 0.9998940229415894)
     (* (cos (* (* t_0 (* 2.0 u2)) t_0)) (sqrt (- (log (- 1.0 u1)))))
     (* (cos (* (* (PI) 2.0) u2)) (pow (* u1 u1) 0.25)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999894023

   1. Initial program 89.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 89.3

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

   4. Applied rewrites 89.3%

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

   if 0.999894023 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 33.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 94.1

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

   5. Applied rewrites 94.1%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f32 N/A

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

      8. metadata-eval 93.5

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

   7. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left({\left(-\left(-u1\right)\right)}^{0.25} \cdot {\left(-\left(-u1\right)\right)}^{0.25}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-pow.f32 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. sqr-neg N/A

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

      9. lower-*.f32 94.1

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

   9. Applied rewrites 94.1%

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

   10. Taylor expanded in u1 around 0

       \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. 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 94.1

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

   12. Applied rewrites 94.1%

       \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998940229415894:\\ \;\;\;\;\cos \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013000000268220901:\\ \;\;\;\;t\_0 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{-\log \left(\sqrt{1 - u1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* (PI) 2.0) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.013000000268220901)
     (* t_0 (pow (* u1 u1) 0.25))
     (* (sqrt 2.0) (sqrt (- (log (sqrt (- 1.0 u1)))))))))

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))) < 0.0130000003

   1. Initial program 36.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 92.6

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

   5. Applied rewrites 92.6%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f32 N/A

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

      8. metadata-eval 92.0

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

   7. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\left({\left(-\left(-u1\right)\right)}^{0.25} \cdot {\left(-\left(-u1\right)\right)}^{0.25}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-pow.f32 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. sqr-neg N/A

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

      9. lower-*.f32 92.7

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

   9. Applied rewrites 92.7%

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

   10. Taylor expanded in u1 around 0

       \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. 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 92.7

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

   12. Applied rewrites 92.7%

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

   if 0.0130000003 < (*.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 90.3%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 84.7

          \[\leadsto \sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{\color{blue}{-0.5}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 84.7%

      \[\leadsto \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{-0.5}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-sqrt.f32 76.2

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

   7. Applied rewrites 76.2%

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

   9. Applied rewrites 78.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013000000268220901:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{-\log \left(\sqrt{1 - u1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{-\log \left(\sqrt{1 - u1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (cos (* (* (PI) 2.0) u2)) (sqrt (- (log (- 1.0 u1)))))
      0.013000000268220901)
   (* (sqrt u1) (cos (* (+ u2 u2) (PI))))
   (* (sqrt 2.0) (sqrt (- (log (sqrt (- 1.0 u1))))))))

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))) < 0.0130000003

   1. Initial program 36.3%

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 92.6

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

   7. Applied rewrites 92.6%

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

   9. Applied rewrites 92.6%

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

   if 0.0130000003 < (*.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 90.3%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 84.7

          \[\leadsto \sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{\color{blue}{-0.5}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 84.7%

      \[\leadsto \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{-0.5}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-sqrt.f32 76.2

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

   7. Applied rewrites 76.2%

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

   9. Applied rewrites 78.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{-\log \left(\sqrt{1 - u1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998940229415894:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* (PI) 2.0) u2))))
   (if (<= (- 1.0 u1) 0.9998940229415894)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* t_0 (pow (* u1 u1) 0.25)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999894023

   1. Initial program 89.3%

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

   if 0.999894023 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 33.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 94.1

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

   5. Applied rewrites 94.1%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f32 N/A

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

      8. metadata-eval 93.5

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

   7. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left({\left(-\left(-u1\right)\right)}^{0.25} \cdot {\left(-\left(-u1\right)\right)}^{0.25}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-pow.f32 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. sqr-neg N/A

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

      9. lower-*.f32 94.1

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

   9. Applied rewrites 94.1%

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

   10. Taylor expanded in u1 around 0

       \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. 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 94.1

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

   12. Applied rewrites 94.1%

       \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998940229415894:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 46.8%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.3

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

7. Applied rewrites 78.3%

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

9. Applied rewrites 78.3%

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

10. Final simplification 78.3%

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

Alternative 6: 64.9% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 47.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.3

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

7. Applied rewrites 78.3%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 67.0%

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

Reproduce

herbie shell --seed 2024332 
(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))))