Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 91.1%

Time: 8.8s

Alternatives: 8

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

Initial Program: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := {\mathsf{PI}\left(\right)}^{0.16666666666666666}\\ \mathbf{if}\;t\_0 \leq 0.00010599999950500205:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\left(\left(t\_1 \cdot u2\right) \cdot t\_1\right) \cdot 2\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (pow (PI) 0.16666666666666666)))
   (if (<= t_0 0.00010599999950500205)
     (* (pow (* u1 u1) 0.25) (sin (* (+ u2 u2) (PI))))
     (*
      (sin (* (* (* (* t_1 u2) t_1) 2.0) (pow (cbrt (PI)) 2.0)))
      (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.06e-4

   1. Initial program 34.4%

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 93.5

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

   7. Applied rewrites 93.5%

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

   9. Applied rewrites 93.6%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.6%

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

   if 1.06e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.3%

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

      1. lift-*.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. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-*.f32 N/A

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

      2. lift-cbrt.f32 N/A

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

      3. pow1/3 N/A

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

      4. sqr-pow N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{6}}} \cdot \left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot u2\right)\right) \cdot 2\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot u2\right)}\right) \cdot 2\right)\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}} \cdot u2\right)\right) \cdot 2\right)\right) \]

      11. metadata-eval 89.4

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{\color{blue}{0.16666666666666666}} \cdot u2\right)\right) \cdot 2\right)\right) \]

   6. Applied rewrites 89.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot u2\right)\right)} \cdot 2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00010599999950500205:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\left(\left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot 2\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.00010599999950500205:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.00010599999950500205)
     (* (pow (* u1 u1) 0.25) (sin (* (+ u2 u2) (PI))))
     (* (sin (* (* (PI) 2.0) u2)) (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.06e-4

   1. Initial program 34.4%

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 93.5

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

   7. Applied rewrites 93.5%

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

   9. Applied rewrites 93.6%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.6%

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

   if 1.06e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.3%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00010599999950500205:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0009399999980814755:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.0009399999980814755)
   (* (pow (* u1 u1) 0.25) (sin (* (+ u2 u2) (PI))))
   (*
    (sqrt (log (sqrt (/ 1.0 (- 1.0 u1)))))
    (* (* (* (sqrt 2.0) (PI)) u2) 2.0))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 9.39999998e-4

   1. Initial program 39.9%

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

   4. Applied rewrites 44.9%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.4%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.4%

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

   if 9.39999998e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 92.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f32 N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower--.f32 78.3

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

   7. Applied rewrites 78.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0009399999980814755:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9990599751472473:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9990599751472473)
   (*
    (sqrt (log (sqrt (/ 1.0 (- 1.0 u1)))))
    (* (* (* (sqrt 2.0) (PI)) u2) 2.0))
   (* (sqrt u1) (sin (* (+ u2 u2) (PI))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999059975

   1. Initial program 92.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. sub-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-/.f32 N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. lower--.f32 78.3

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

   7. Applied rewrites 78.3%

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

   if 0.999059975 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 39.9%

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9990599751472473:\\ \;\;\;\;\sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)} \cdot \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.2

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

7. Applied rewrites 78.2%

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

9. Applied rewrites 78.2%

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

10. Final simplification 78.2%

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

Alternative 6: 66.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.2

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

7. Applied rewrites 78.2%

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

9. Applied rewrites 78.2%

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

10. Taylor expanded in u2 around 0

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

12. Applied rewrites 68.6%

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

13. Final simplification 68.6%

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

Alternative 7: 66.3% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 37.4%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.2

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

7. Applied rewrites 78.2%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 68.6%

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

11. Final simplification 68.6%

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

Alternative 8: 66.3% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.4%

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

4. Applied rewrites 35.9%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.2

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

7. Applied rewrites 78.2%

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

9. Applied rewrites 78.2%

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

10. Taylor expanded in u2 around 0

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

12. Applied rewrites 68.5%

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

Reproduce

herbie shell --seed 2024332 
(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))))