Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 91.2%

Time: 9.7s

Alternatives: 6

Speedup: 9.6×

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: 57.7% 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: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := {t\_0}^{2}\\ t_2 := u2 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sin \left(\left(\left(\left(u2 \cdot t\_1\right) \cdot 2\right) \cdot \sqrt[3]{t\_0}\right) \cdot \sqrt[3]{t\_1}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\cos t\_2 \cdot \sin t\_2\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))) (t_1 (pow t_0 2.0)) (t_2 (* u2 (PI))))
   (if (<= (- 1.0 u1) 0.9998400211334229)
     (*
      (sin (* (* (* (* u2 t_1) 2.0) (cbrt t_0)) (cbrt t_1)))
      (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) (* (* (cos t_2) (sin t_2)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999840021

   1. Initial program 89.7%

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

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

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. pow2 N/A

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

      14. lower-pow.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lower-cbrt.f32 89.6

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

   4. Applied rewrites 89.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. rem-cbrt-cube N/A

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

      4. cube-unmult N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f32 N/A

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

      7. cbrt-prod N/A

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

      8. pow1/3 N/A

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

      9. pow1/3 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f32 N/A

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

      13. pow1/3 N/A

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

      14. lower-cbrt.f32 N/A

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

      15. lower-*.f32 N/A

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

   6. Applied rewrites 89.7%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 39.4%

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

   4. Applied rewrites 39.9%

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

   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. lower-sqrt.f32 91.9

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 91.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sin \left(\left(\left(\left(u2 \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot 2\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.00015999999595806003:\\ \;\;\;\;t\_1 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sin (* (* 2.0 (PI)) u2))))
   (if (<= t_0 0.00015999999595806003) (* t_1 (sqrt u1)) (* t_1 (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.59999996e-4

   1. Initial program 39.4%

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

   4. Applied rewrites 42.2%

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

   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. lower-sqrt.f32 91.9

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

   7. Applied rewrites 91.9%

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

   if 1.59999996e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00015999999595806003:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\cos t\_0 \cdot \sin t\_0\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))))
   (if (<= (- 1.0 u1) 0.9998400211334229)
     (* (sin (* (* 2.0 (PI)) u2)) (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) (* (* (cos t_0) (sin t_0)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999840021

   1. Initial program 89.7%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 39.4%

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

   4. Applied rewrites 41.2%

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

   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. lower-sqrt.f32 91.9

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 91.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\cos \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9800000190734863)
   (*
    (* (* (fma (* -1.3333333333333333 (* u2 u2)) (* (PI) (PI)) 2.0) (PI)) u2)
    (sqrt (- (log (- 1.0 u1)))))
   (* (sin (* (* 2.0 (PI)) u2)) (sqrt u1))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.980000019

   1. Initial program 97.0%

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

   3. 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 86.6%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right)} \]

   if 0.980000019 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 53.4%

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

   4. Applied rewrites 39.3%

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

   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 N/A

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

      16. lower-sqrt.f32 82.4

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

   7. Applied rewrites 82.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.8%

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

4. Applied rewrites 37.6%

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

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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-sqrt.f32 73.3

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

7. Applied rewrites 73.3%

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

8. Final simplification 73.3%

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

Alternative 6: 66.5% accurate, 9.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.8%

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

4. Applied rewrites 37.8%

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

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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. lower-sqrt.f32 73.3

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

7. Applied rewrites 73.3%

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

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 63.9%

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

12. Applied rewrites 63.9%

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

13. Final simplification 63.9%

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

Reproduce

herbie shell --seed 2024295 
(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))))