Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.1% → 98.3%

Time: 10.7s

Alternatives: 9

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

Initial Program: 58.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 56.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. *-lft-identity N/A

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

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

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

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

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.1899999976158142:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \left(2 \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.1899999976158142)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))
   (*
    (* u2 (* 2.0 (* (sqrt 2.0) (PI))))
    (sqrt (log (sqrt (/ 1.0 (- 1.0 u1))))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.189999998

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 97.0

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

   5. Applied rewrites 97.0%

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

   if 0.189999998 < u1

   1. Initial program 98.5%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

   7. Applied rewrites 89.0%

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

Alternative 3: 93.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
  (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 92.6

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

5. Applied rewrites 92.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Add Preprocessing

Alternative 4: 91.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ \mathbf{if}\;u2 \leq 0.00019999999494757503:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, 0.16666666666666666 \cdot t\_0\right), u1, t\_0 \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))))
   (if (<= u2 0.00019999999494757503)
     (*
      (fma
       (fma
        (fma
         (* (- 0.25 (/ 0.0625 u1)) (sqrt u1))
         0.5
         (* 0.16666666666666666 t_0))
        u1
        (* t_0 0.25))
       (* u1 u1)
       (sqrt u1))
      (* (* (PI) u2) 2.0))
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if u2 < 1.99999995e-4

   1. Initial program 58.5%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. metadata-eval 98.0

          \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 76.3

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

   10. Applied rewrites 76.3%

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

   11. Taylor expanded in u1 around 0

       \[\leadsto \color{blue}{\left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) \cdot {u1}^{2}} + \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       3. lower-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right), {u1}^{2}, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

   13. Applied rewrites 92.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, 0.16666666666666666 \cdot \sqrt{\frac{1}{u1}}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

   if 1.99999995e-4 < u2

   1. Initial program 53.4%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 5: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
  (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 90.7

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

5. Applied rewrites 90.7%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Add Preprocessing

Alternative 6: 87.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ \mathbf{if}\;u2 \leq 0.001500000013038516:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, 0.16666666666666666 \cdot t\_0\right), u1, t\_0 \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))))
   (if (<= u2 0.001500000013038516)
     (*
      (fma
       (fma
        (fma
         (* (- 0.25 (/ 0.0625 u1)) (sqrt u1))
         0.5
         (* 0.16666666666666666 t_0))
        u1
        (* t_0 0.25))
       (* u1 u1)
       (sqrt u1))
      (* (* (PI) u2) 2.0))
     (* (sqrt u1) (sin (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00150000001

   1. Initial program 57.5%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. metadata-eval 98.0

          \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.4

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

   7. Applied rewrites 76.4%

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

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 75.4

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

   10. Applied rewrites 75.4%

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

   11. Taylor expanded in u1 around 0

       \[\leadsto \color{blue}{\left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) + \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right) \cdot {u1}^{2}} + \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       3. lower-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right), {u1}^{2}, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

   13. Applied rewrites 90.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, 0.16666666666666666 \cdot \sqrt{\frac{1}{u1}}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

   if 0.00150000001 < u2

   1. Initial program 54.2%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. metadata-eval 97.4

          \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 7: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0010999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0010999999940395355)
   (*
    (fma
     (fma (sqrt (/ 1.0 u1)) 0.25 (* 0.16666666666666666 (sqrt u1)))
     (* u1 u1)
     (sqrt u1))
    (* (* (PI) u2) 2.0))
   (* (sqrt u1) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0011

   1. Initial program 57.8%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. metadata-eval 98.0

          \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 75.8

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

   10. Applied rewrites 75.8%

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

   11. Taylor expanded in u1 around 0

       \[\leadsto \color{blue}{\left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}\right) + \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}\right) \cdot {u1}^{2}} + \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       3. lower-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}, {u1}^{2}, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{6} \cdot \sqrt{u1}}, {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{u1}} \cdot \frac{1}{4}} + \frac{1}{6} \cdot \sqrt{u1}, {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       6. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right)}, {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       7. lower-sqrt.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{u1}}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       8. lower-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{u1}}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       9. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \color{blue}{\frac{1}{6} \cdot \sqrt{u1}}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       10. lower-sqrt.f32 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \color{blue}{\sqrt{u1}}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), \color{blue}{u1 \cdot u1}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       12. lower-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), \color{blue}{u1 \cdot u1}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

       13. lower-sqrt.f32 89.0

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \color{blue}{\sqrt{u1}}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

   13. Applied rewrites 89.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

   if 0.0011 < u2

   1. Initial program 53.7%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-log.f32 N/A

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

      8. lift--.f32 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. metadata-eval 97.4

          \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 97.4%

      \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 77.0

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

   7. Applied rewrites 77.0%

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

Alternative 8: 76.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma
   (fma (sqrt (/ 1.0 u1)) 0.25 (* 0.16666666666666666 (sqrt u1)))
   (* u1 u1)
   (sqrt u1))
  (* (* (PI) u2) 2.0)))

Derivation

1. Initial program 56.5%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift--.f32 N/A

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

   9. *-lft-identity N/A

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

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

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. metadata-eval 97.8

       \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 97.8%

   \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.5

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

7. Applied rewrites 76.5%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 65.4

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

10. Applied rewrites 65.4%

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

11. Taylor expanded in u1 around 0

    \[\leadsto \color{blue}{\left(\sqrt{u1} + {u1}^{2} \cdot \left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}\right) + \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    2. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}\right) \cdot {u1}^{2}} + \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    3. lower-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{u1} + \frac{1}{4} \cdot \sqrt{\frac{1}{u1}}, {u1}^{2}, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{6} \cdot \sqrt{u1}}, {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{u1}} \cdot \frac{1}{4}} + \frac{1}{6} \cdot \sqrt{u1}, {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    6. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right)}, {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    7. lower-sqrt.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{u1}}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    8. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{u1}}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    9. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \color{blue}{\frac{1}{6} \cdot \sqrt{u1}}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    10. lower-sqrt.f32 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \color{blue}{\sqrt{u1}}\right), {u1}^{2}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), \color{blue}{u1 \cdot u1}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    12. lower-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, \frac{1}{4}, \frac{1}{6} \cdot \sqrt{u1}\right), \color{blue}{u1 \cdot u1}, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

    13. lower-sqrt.f32 74.9

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \color{blue}{\sqrt{u1}}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]

13. Applied rewrites 74.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \sqrt{u1}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]
14. Add Preprocessing

Alternative 9: 66.3% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.5%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lift-log.f32 N/A

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

   8. lift--.f32 N/A

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

   9. *-lft-identity N/A

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

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

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. metadata-eval 97.8

       \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 97.8%

   \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.5

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

7. Applied rewrites 76.5%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 65.4

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

10. Applied rewrites 65.4%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))