Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.3%

Time: 11.2s

Alternatives: 12

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: 57.8% 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, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

   1. lift-*.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. lift-sin.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-cos.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. sin-cos-mult N/A

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

   7. lower-/.f32 N/A

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

6. Applied rewrites 98.6%

   \[\leadsto \left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 2\right) \cdot \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot u2 - \left(-\mathsf{PI}\left(\right)\right) \cdot u2\right) + \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), u2, \left(-\mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}{2}} \]
7. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

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

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

   5. lift-*.f32 N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 98.6

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

8. Applied rewrites 98.6%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03799999877810478)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (* (PI) (+ u2 u2))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0379999988

   1. Initial program 47.8%

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

   3. 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 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.0379999988 < u1

   1. Initial program 97.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. count-2-rev N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f32 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-cbrt.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lower-cbrt.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-fma.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-pow.f32 N/A

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

      9. unpow2 N/A

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

      10. cube-unmult N/A

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

      11. lift-cbrt.f32 N/A

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

      12. rem-cube-cbrt N/A

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

      13. distribute-lft-out N/A

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

      14. lower-*.f32 N/A

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

      15. lower-+.f32 97.5

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

   6. Applied rewrites 97.5%

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

Alternative 3: 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 55.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 4: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03799999877810478)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))
   (*
    (sqrt (- (log (- 1.0 u1))))
    (*
     (+
      (fma (* (* u2 u2) -1.3333333333333333) (* (* (PI) (PI)) (PI)) (PI))
      (PI))
     u2))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0379999988

   1. Initial program 47.8%

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

   3. 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 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.0379999988 < u1

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

   5. Applied rewrites 90.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00022000000171829015:\\ \;\;\;\;\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 2\right) \cdot \frac{\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00022000000171829015)
   (* (* (sqrt (- (log1p (- u1)))) 2.0) (/ (* (* (PI) u2) 2.0) 2.0))
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 2.20000002e-4

   1. Initial program 56.4%

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

      1. lift-*.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. lift-sin.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f32 N/A

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

      4. lift-cos.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f32 N/A

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 98.4

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

   9. Applied rewrites 98.4%

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

   if 2.20000002e-4 < u2

   1. Initial program 53.9%

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

   3. 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 93.7

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

   5. Applied rewrites 93.7%

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

Alternative 6: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00022000000171829015:\\ \;\;\;\;\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 2\right) \cdot \frac{\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00022000000171829015)
   (* (* (sqrt (- (log1p (- u1)))) 2.0) (/ (* (* (PI) u2) 2.0) 2.0))
   (*
    (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 2.20000002e-4

   1. Initial program 56.4%

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

      1. lift-*.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. lift-sin.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f32 N/A

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

      4. lift-cos.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. sin-cos-mult N/A

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

      7. lower-/.f32 N/A

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 98.4

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

   9. Applied rewrites 98.4%

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

   if 2.20000002e-4 < u2

   1. Initial program 53.9%

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

   3. 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 92.3

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

   5. Applied rewrites 92.3%

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

Alternative 7: 91.8% 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 55.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 91.6

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

5. Applied rewrites 91.6%

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

Alternative 8: 88.2% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 88.7

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

5. Applied rewrites 88.7%

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

Alternative 9: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ \mathbf{if}\;u2 \leq 0.0012499999720603228:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, 0.16666666666666666 \cdot t\_0\right), u1, t\_0 \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))))
   (if (<= u2 0.0012499999720603228)
     (*
      (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 < 0.00124999997

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 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 76.9

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

   7. Applied rewrites 76.9%

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

   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 76.6

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

   10. Applied rewrites 76.6%

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

   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 92.0%

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

   if 0.00124999997 < u2

   1. Initial program 50.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 97.6

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

   4. Applied rewrites 97.6%

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

   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 82.6

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

   7. Applied rewrites 82.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0012499999720603228:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, 0.16666666666666666 \cdot \sqrt{\frac{1}{u1}}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0012499999720603228:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0012499999720603228)
   (*
    (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 < 0.00124999997

   1. Initial program 57.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 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 76.9

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

   7. Applied rewrites 76.9%

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

   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 76.6

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

   10. Applied rewrites 76.6%

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

   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 89.7

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

   13. Applied rewrites 89.7%

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

   if 0.00124999997 < u2

   1. Initial program 50.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 97.6

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

   4. Applied rewrites 97.6%

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

   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 82.6

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

   7. Applied rewrites 82.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0012499999720603228:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{u1}}, 0.25, 0.16666666666666666 \cdot \sqrt{u1}\right), u1 \cdot u1, \sqrt{u1}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 55.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 97.9

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

4. Applied rewrites 97.9%

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

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 78.5

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

7. Applied rewrites 78.5%

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

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 67.7

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

10. Applied rewrites 67.7%

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

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 77.4

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

13. Applied rewrites 77.4%

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

14. Final simplification 77.4%

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

Alternative 12: 66.5% 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 55.5%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 97.9

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

4. Applied rewrites 97.9%

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

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 78.5

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

7. Applied rewrites 78.5%

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

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 67.7

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

10. Applied rewrites 67.7%

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

11. Final simplification 67.7%

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

Reproduce

herbie shell --seed 2024350 
(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))))