Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 98.1%

Time: 11.3s

Alternatives: 9

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

Initial Program: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - u1 \cdot u1\\ t_1 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.03200000151991844:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log t\_0 - \log \left(t\_0 \cdot \left(1 - u1\right)\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- 1.0 (* u1 u1))) (t_1 (sin (* (* 2.0 (PI)) u2))))
   (if (<= (- (log (- 1.0 u1))) 0.03200000151991844)
     (*
      (sqrt
       (-
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      t_1)
     (* (sqrt (- (log t_0) (log (* t_0 (- 1.0 u1))))) t_1))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0320000015

   1. Initial program 50.3%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right)} \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1} - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1} - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4} \cdot u1\right)\right)} - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower--.f32 N/A

          \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot u1\right)\right) - \frac{1}{3}\right)} \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot u1} - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\frac{-1}{4}} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. lower-*.f32 98.2

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{-0.25 \cdot u1} - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.0320000015 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 97.2%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lift--.f32 N/A

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. flip-+ N/A

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

      8. lift--.f32 N/A

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

      9. frac-times N/A

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

      10. log-div N/A

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

      11. lower--.f32 N/A

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

      12. lower-log.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-log.f32 N/A

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

   4. Applied rewrites 97.8%

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

      1. lift-*.f32 N/A

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

      2. *-lft-identity 97.8

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

   6. Applied rewrites 97.8%

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

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sin (* (* 2.0 (PI)) u2))))
   (if (<= t_0 0.03999999910593033)
     (*
      (sqrt
       (-
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      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))) < 0.0399999991

   1. Initial program 50.5%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right)} \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1} - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1} - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4} \cdot u1\right)\right)} - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower--.f32 N/A

          \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot u1\right)\right) - \frac{1}{3}\right)} \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot u1} - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\frac{-1}{4}} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. lower-*.f32 98.2

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{-0.25 \cdot u1} - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.0399999991 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 97.4%

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

Alternative 3: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9480000138282776:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9480000138282776)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log (- 1.0 u1)))))
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.948000014

   1. Initial program 97.6%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 37.5

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

   7. Applied rewrites 37.5%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower--.f32 87.3

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

   10. Applied rewrites 87.3%

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

   if 0.948000014 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 51.1%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot u1} - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower--.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right)} \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1} - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1} - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4} \cdot u1\right)\right)} - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower--.f32 N/A

          \[\leadsto \sqrt{-\left(\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot u1\right)\right) - \frac{1}{3}\right)} \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot u1} - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{\frac{-1}{4}} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. lower-*.f32 98.1

          \[\leadsto \sqrt{-\left(\left(\left(\color{blue}{-0.25 \cdot u1} - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9750000238418579:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9750000238418579)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log (- 1.0 u1)))))
   (*
    (sqrt (- (* (- (* (- (* -0.3333333333333333 u1) 0.5) u1) 1.0) u1)))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.975000024

   1. Initial program 97.0%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 39.5

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower--.f32 86.6

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

   10. Applied rewrites 86.6%

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

   if 0.975000024 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 49.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower--.f32 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f32 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 5: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9879999756813049:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9879999756813049)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log (- 1.0 u1)))))
   (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.987999976

   1. Initial program 95.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 42.3

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

   7. Applied rewrites 42.3%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower--.f32 84.0

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

   10. Applied rewrites 84.0%

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

   if 0.987999976 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 47.3%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower--.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f32 96.8

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

   5. Applied rewrites 96.8%

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

Alternative 6: 85.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9980999827384949:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9980999827384949)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log (- 1.0 u1)))))
   (* (sqrt u1) (sin (* (PI) (+ u2 u2))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998099983

   1. Initial program 93.6%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 92.5

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 46.0

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower--.f32 79.4

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

   10. Applied rewrites 79.4%

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

   if 0.998099983 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 43.9%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 41.0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 88.2

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

   7. Applied rewrites 88.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 88.2

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

   9. Applied rewrites 88.2%

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

Alternative 7: 76.1% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 53.9

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

4. Applied rewrites 53.9%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 77.7

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

9. Applied rewrites 77.7%

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

Alternative 8: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(u1 \cdot u1\right)}^{0.25} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (* u1 u1) 0.25) (* (* (PI) u2) 2.0)))

Derivation

1. Initial program 56.3%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 53.9

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

4. Applied rewrites 53.9%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 66.6

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

10. Applied rewrites 66.6%

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

12. Applied rewrites 66.6%

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

Alternative 9: 66.1% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 53.9

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

4. Applied rewrites 53.9%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 66.6

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

10. Applied rewrites 66.6%

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

Reproduce

herbie shell --seed 2024333 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))