Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 91.3%

Time: 9.0s

Alternatives: 5

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: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998499751091003)
   (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2)))
   (* (sin (* (* (PI) 2.0) u2)) (sqrt u1))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999849975

   1. Initial program 88.4%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 37.5%

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

   4. Applied rewrites 40.3%

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

   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 93.2

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

   7. Applied rewrites 93.2%

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

Alternative 2: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0003209999995306134:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.0003209999995306134)
   (* (sin (* (* (PI) 2.0) u2)) (sqrt u1))
   (*
    (* 2.0 (* (* (sqrt 2.0) (PI)) u2))
    (sqrt (log (sqrt (/ 1.0 (- 1.0 u1))))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 3.21e-4

   1. Initial program 39.8%

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

   4. Applied rewrites 38.8%

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

   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 91.7

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

   7. Applied rewrites 91.7%

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

   if 3.21e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. 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 85.1

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

   4. Applied rewrites 85.1%

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

   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. lower-/.f32 N/A

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

      14. lower--.f32 73.4

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

   7. Applied rewrites 73.4%

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

Alternative 3: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.019999999552965164:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.019999999552965164)
     (* (sin (* (* (PI) 2.0) u2)) (sqrt u1))
     (*
      (sqrt t_0)
      (*
       (* (PI) (fma (* (* u2 u2) -1.3333333333333333) (* (PI) (PI)) 2.0))
       u2)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0199999996

   1. Initial program 52.9%

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

   4. Applied rewrites 36.4%

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

   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%

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

   if 0.0199999996 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 96.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. 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 80.2%

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

Alternative 4: 76.1% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.6%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 76.1

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

7. Applied rewrites 76.1%

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

Alternative 5: 66.0% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.6%

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

4. Applied rewrites 14.5%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower-log1p.f32 67.4

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

7. Applied rewrites 67.4%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 67.8%

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

Reproduce

herbie shell --seed 2024311 
(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))))