Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.5% → 91.4%

Time: 8.6s

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: 57.5% 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: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \end{array} \]

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 0.0001500000071246177)
     (* (sqrt (/ (* (* (- (- u1) 1.0) u1) 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))) < 1.50000007e-4

   1. Initial program 33.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 94.0

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

   5. Applied rewrites 94.0%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 94.0

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 94.0

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

   10. Applied rewrites 94.0%

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

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 94.0

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

   13. Applied rewrites 94.0%

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

   if 1.50000007e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   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. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.017999999225139618)
     (* (sqrt (/ (* (* (- (- u1) 1.0) u1) u1) (- u1))) t_0)
     (* (sqrt (log (sqrt (/ 1.0 (- 1.0 u1))))) (sqrt 2.0)))))

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))) < 0.0179999992

   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 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 92.1

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

   10. Applied rewrites 92.1%

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

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 92.1

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

   13. Applied rewrites 92.1%

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

   if 0.0179999992 < (*.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 91.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 86.7

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

   4. Applied rewrites 86.7%

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

   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 75.8

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

   7. Applied rewrites 75.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{\frac{\left(-u1\right) \cdot u1}{-u1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.017999999225139618)
     (* (sqrt (/ (* (- u1) u1) (- u1))) t_0)
     (* (sqrt (log (sqrt (/ 1.0 (- 1.0 u1))))) (sqrt 2.0)))))

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))) < 0.0179999992

   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 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{-1 \cdot {u1}^{2}}}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f32 92.0

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

   10. Applied rewrites 92.0%

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

   if 0.0179999992 < (*.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 91.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 86.7

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

   4. Applied rewrites 86.7%

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

   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 75.8

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

   7. Applied rewrites 75.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{\frac{\left(-u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1 - \left(-u1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (* (fma -0.5 u1 1.0) u1) (* (- u1) u1)))
  (cos (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 54.1%

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

4. Applied rewrites 49.6%

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

5. Taylor expanded in u1 around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 53.2

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

7. Applied rewrites 52.0%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 61.0

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

10. Applied rewrites 60.7%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \left(-u1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
11. Add Preprocessing

Alternative 5: 76.4% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 54.1%

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

4. Applied rewrites 48.6%

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

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 78.7

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

7. Applied rewrites 78.7%

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

Alternative 6: 64.7% accurate, 14.4× speedup

Math

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

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(u1) * Float32(1.0))
end

MATLAB

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

Derivation

1. Initial program 54.1%

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

4. Applied rewrites 46.6%

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

5. Taylor expanded in u2 around 0

   \[\leadsto e^{\log \left(\mathsf{log1p}\left(u1\right)\right) \cdot \frac{1}{2}} \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 8.3%

   \[\leadsto e^{\log \left(\mathsf{log1p}\left(u1\right)\right) \cdot 0.5} \cdot \color{blue}{1} \]

8. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 66.5

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]

10. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2024302 
(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))))