Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 98.6%

Time: 12.4s

Alternatives: 15

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: 57.3% 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: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;{\left(\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(\left(\frac{\left(0.3333333333333333 + \frac{1}{u1 \cdot u1}\right) + \frac{0.5}{u1}}{-u1} - 0.25\right) \cdot {u1}^{4}\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.030500000342726707)
   (*
    (pow
     (*
      (* (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1)
      (*
       (-
        (/ (+ (+ 0.3333333333333333 (/ 1.0 (* u1 u1))) (/ 0.5 u1)) (- u1))
        0.25)
       (pow u1 4.0)))
     0.25)
    (cos (* (* 2.0 (PI)) u2)))
   (* (sqrt (/ -1.0 (/ 1.0 (log (- 1.0 u1))))) (cos (* -2.0 (* (PI) u2))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0305000003

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. metadata-eval 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in u1 around -inf

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

   10. Applied rewrites 98.8%

       \[\leadsto {\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\frac{\left(0.3333333333333333 + \frac{1}{u1 \cdot u1}\right) + \frac{0.5}{u1}}{-u1} - 0.25\right) \cdot \color{blue}{{u1}^{4}}\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.0305000003 < u1

   1. Initial program 97.1%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-log.f32 97.0

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f32 N/A

          \[\leadsto {\left(e^{\log \left(-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)}\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-neg.f32 25.8

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

   4. Applied rewrites 25.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-log.f32 38.7

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

   7. Applied rewrites 38.7%

      \[\leadsto {\left(e^{\color{blue}{\log u1}}\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

   9. Applied rewrites 38.7%

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

   10. Taylor expanded in u2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower-sqrt.f32 N/A

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

       4. lower-/.f32 N/A

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

       5. exp-neg N/A

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

       6. lower-/.f32 N/A

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

       7. rem-exp-log N/A

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

       8. lower-neg.f32 N/A

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

       9. lower-log.f32 N/A

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

       10. lower--.f32 N/A

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

       11. cos-neg-rev N/A

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

       12. lower-cos.f32 N/A

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

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

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

       14. metadata-eval N/A

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

       15. lower-*.f32 N/A

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

       16. *-commutative N/A

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

   12. Applied rewrites 97.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;{\left(\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(\left(\frac{\left(0.3333333333333333 + \frac{1}{u1 \cdot u1}\right) + \frac{0.5}{u1}}{-u1} - 0.25\right) \cdot {u1}^{4}\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.17499999701976776:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(\left(-2 \cdot \left(u2 \cdot u2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.17499999701976776)
     (*
      (sqrt
       (-
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      t_1)
     (* t_0 (+ 1.0 (* (* (* -2.0 (* u2 u2)) (PI)) (PI)))))))

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))) < 0.174999997

   1. Initial program 53.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.174999997 < (*.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 97.4%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 86.5

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

   5. Applied rewrites 86.5%

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

   7. Applied rewrites 92.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.17499999701976776:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \left(\left(-2 \cdot \left(u2 \cdot u2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(\left(-2 \cdot \left(u2 \cdot u2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.11999999731779099)
     (*
      (sqrt (- (* (- (* (- (* -0.3333333333333333 u1) 0.5) u1) 1.0) u1)))
      t_1)
     (* t_0 (+ 1.0 (* (* (* -2.0 (* u2 u2)) (PI)) (PI)))))))

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))) < 0.119999997

   1. Initial program 51.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lower--.f32 N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. metadata-eval N/A

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

      15. lower-*.f32 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.119999997 < (*.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 97.1%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 93.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{-\left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \left(\left(-2 \cdot \left(u2 \cdot u2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.07599999755620956:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 + \left(\left(-2 \cdot \left(u2 \cdot u2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.07599999755620956)
     (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) t_1)
     (* t_0 (+ 1.0 (* (* (* -2.0 (* u2 u2)) (PI)) (PI)))))))

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))) < 0.0759999976

   1. Initial program 50.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. lower--.f32 N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f32 96.4

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

   5. Applied rewrites 96.4%

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

   if 0.0759999976 < (*.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 96.0%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 77.2

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 92.6%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.07599999755620956:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \left(\left(-2 \cdot \left(u2 \cdot u2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-t\_0} \cdot t\_1 \leq 0.08500000089406967:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{t\_0}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- t_0)) t_1) 0.08500000089406967)
     (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) t_1)
     (sqrt (/ -1.0 (/ 1.0 t_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))) < 0.0850000009

   1. Initial program 51.4%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. lower--.f32 N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f32 96.1

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

   5. Applied rewrites 96.1%

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

   if 0.0850000009 < (*.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 96.7%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-log.f32 96.7

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f32 N/A

          \[\leadsto {\left(e^{\log \left(-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)}\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-neg.f32 28.2

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

   4. Applied rewrites 28.2%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-log.f32 41.4

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

   7. Applied rewrites 41.4%

      \[\leadsto {\left(e^{\color{blue}{\log u1}}\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

   9. Applied rewrites 41.4%

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

   10. Taylor expanded in u2 around 0

       \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\mathsf{neg}\left(\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)\right)}}}} \]
   11. Step-by-step derivation

       1. lower-sqrt.f32 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\mathsf{neg}\left(\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)\right)}}}} \]

       2. lower-/.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)\right)}}}} \]

       3. exp-neg N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{e^{\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)}}}}} \]

       4. lower-/.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{e^{\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)}}}}} \]

       5. rem-exp-log N/A

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

       6. lower-neg.f32 N/A

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

       7. lower-log.f32 N/A

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

       8. lower--.f32 87.0

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

   12. Applied rewrites 87.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.08500000089406967:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-t\_0} \cdot t\_1 \leq 0.019600000232458115:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{t\_0}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- t_0)) t_1) 0.019600000232458115)
     (* (sqrt u1) t_1)
     (sqrt (/ -1.0 (/ 1.0 t_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))) < 0.0196000002

   1. Initial program 45.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 44.8

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 87.2

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

   7. Applied rewrites 87.2%

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

   if 0.0196000002 < (*.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 91.1%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-log.f32 91.1

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f32 N/A

          \[\leadsto {\left(e^{\log \left(-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)}\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-neg.f32 33.4

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

   4. Applied rewrites 33.6%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-log.f32 49.1

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

   7. Applied rewrites 49.1%

      \[\leadsto {\left(e^{\color{blue}{\log u1}}\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

   9. Applied rewrites 49.1%

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

   10. Taylor expanded in u2 around 0

       \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\mathsf{neg}\left(\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)\right)}}}} \]
   11. Step-by-step derivation

       1. lower-sqrt.f32 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{e^{\mathsf{neg}\left(\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)\right)}}}} \]

       2. lower-/.f32 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)\right)}}}} \]

       3. exp-neg N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{e^{\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)}}}}} \]

       4. lower-/.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{e^{\log \left(\mathsf{neg}\left(\log \left(1 - u1\right)\right)\right)}}}}} \]

       5. rem-exp-log N/A

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

       6. lower-neg.f32 N/A

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

       7. lower-log.f32 N/A

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

       8. lower--.f32 81.0

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

   12. Applied rewrites 81.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.019600000232458115:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.019600000232458115:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.019600000232458115) (* (sqrt u1) t_1) (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0196000002

   1. Initial program 45.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 44.8

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 87.2

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

   7. Applied rewrites 87.2%

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

   if 0.0196000002 < (*.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 91.1%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 80.9%

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

Alternative 8: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.010099999606609344:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.010099999606609344)
     (* (pow (* u1 u1) 0.25) 1.0)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0100999996

   1. Initial program 42.1%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 46.8

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in u2 around 0

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

   10. Applied rewrites 68.7%

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

   12. Applied rewrites 68.8%

       \[\leadsto {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \cdot 1 \]

   if 0.0100999996 < (*.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 88.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 77.0%

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

Alternative 9: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;{\left({\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)}^{4}\right)}^{0.125} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.030500000342726707)
   (*
    (pow
     (pow
      (* (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1)
      4.0)
     0.125)
    (cos (* (* 2.0 (PI)) u2)))
   (* (sqrt (/ -1.0 (/ 1.0 (log (- 1.0 u1))))) (cos (* -2.0 (* (PI) u2))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0305000003

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. metadata-eval 98.7

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

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{{\left(\left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{0.25}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-pow.f32 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(\left(-\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1\right) \cdot \left(-\left(\left(\left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) \cdot u1 - \frac{1}{2}\right) \cdot u1 - 1\right) \cdot u1\right)\right)}^{\color{blue}{\left(2 \cdot \frac{1}{8}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

   9. Applied rewrites 98.7%

      \[\leadsto \color{blue}{{\left({\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)}^{4}\right)}^{0.125}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.0305000003 < u1

   1. Initial program 97.1%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-log.f32 97.0

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f32 N/A

          \[\leadsto {\left(e^{\log \left(-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)}\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-neg.f32 25.6

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

   4. Applied rewrites 25.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-log.f32 38.7

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

   7. Applied rewrites 38.7%

      \[\leadsto {\left(e^{\color{blue}{\log u1}}\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

   9. Applied rewrites 38.7%

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

   10. Taylor expanded in u2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower-sqrt.f32 N/A

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

       4. lower-/.f32 N/A

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

       5. exp-neg N/A

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

       6. lower-/.f32 N/A

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

       7. rem-exp-log N/A

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

       8. lower-neg.f32 N/A

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

       9. lower-log.f32 N/A

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

       10. lower--.f32 N/A

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

       11. cos-neg-rev N/A

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

       12. lower-cos.f32 N/A

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

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

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

       14. metadata-eval N/A

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

       15. lower-*.f32 N/A

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

       16. *-commutative N/A

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

   12. Applied rewrites 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;{\left({\left(\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1\right)}^{4}\right)}^{0.125} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.030500000342726707)
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (cos (* (* 2.0 (PI)) u2)))
   (* (sqrt (/ -1.0 (/ 1.0 (log (- 1.0 u1))))) (cos (* -2.0 (* (PI) u2))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0305000003

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0305000003 < u1

   1. Initial program 97.1%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-exp.f32 N/A

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

      7. lower-log.f32 97.0

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

      8. lift-log.f32 N/A

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

      9. lift--.f32 N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f32 N/A

          \[\leadsto {\left(e^{\log \left(-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)}\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      15. lower-neg.f32 26.1

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

   4. Applied rewrites 25.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-log.f32 38.7

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

   7. Applied rewrites 38.7%

      \[\leadsto {\left(e^{\color{blue}{\log u1}}\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-exp.f32 N/A

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

      2. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

   9. Applied rewrites 38.7%

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

   10. Taylor expanded in u2 around inf

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower-sqrt.f32 N/A

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

       4. lower-/.f32 N/A

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

       5. exp-neg N/A

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

       6. lower-/.f32 N/A

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

       7. rem-exp-log N/A

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

       8. lower-neg.f32 N/A

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

       9. lower-log.f32 N/A

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

       10. lower--.f32 N/A

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

       11. cos-neg-rev N/A

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

       12. lower-cos.f32 N/A

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

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

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

       14. metadata-eval N/A

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

       15. lower-*.f32 N/A

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

       16. *-commutative N/A

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

   12. Applied rewrites 97.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\log \left(1 - u1\right)}}} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= u1 0.03500000014901161)
     (*
      (sqrt
       (-
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      t_0)
     (* (sqrt (log (/ 1.0 (- 1.0 u1)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0350000001

   1. Initial program 53.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0350000001 < u1

   1. Initial program 97.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-log.f32 N/A

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

      5. lower-/.f32 97.1

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

   4. Applied rewrites 97.1%

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

Alternative 12: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.030500000342726707:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= u1 0.030500000342726707)
     (*
      (sqrt
       (-
        (*
         (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
         u1)))
      t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0305000003

   1. Initial program 52.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0305000003 < u1

   1. Initial program 97.1%

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

Alternative 13: 65.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ {\left(u1 \cdot u1\right)}^{0.25} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (pow (* u1 u1) 0.25) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 * u1), 0.25f) * 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 = ((u1 * u1) ** 0.25e0) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(u1 * u1) ^ Float32(0.25)) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((u1 * u1) ^ single(0.25)) * single(1.0);
end

Derivation

1. Initial program 60.2%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. lower-log1p.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log1p.f32 42.6

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

4. Applied rewrites 43.2%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 75.2

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

7. Applied rewrites 75.2%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 61.6%

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

12. Applied rewrites 61.7%

    \[\leadsto {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \cdot 1 \]
13. Add Preprocessing

Alternative 14: 65.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ {u1}^{0.5} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (pow u1 0.5) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(u1, 0.5f) * 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 = (u1 ** 0.5e0) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((u1 ^ Float32(0.5)) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u1 ^ single(0.5)) * single(1.0);
end

Derivation

1. Initial program 60.2%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. lower-log1p.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log1p.f32 43.6

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

4. Applied rewrites 43.2%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 75.2

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

7. Applied rewrites 75.2%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 61.6%

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

12. Applied rewrites 61.6%

    \[\leadsto {u1}^{\color{blue}{0.5}} \cdot 1 \]
13. Add Preprocessing

Alternative 15: 65.3% 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 60.2%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. lower-log1p.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log1p.f32 42.4

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 75.2

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

7. Applied rewrites 75.2%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 61.6%

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

Reproduce

herbie shell --seed 2024337 
(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))))