Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.7%

Time: 10.7s

Alternatives: 4

Speedup: 1.1×

Specification

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt (PI))) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt (PI))) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))

Alternative 1: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\sqrt{\left(\left(1 - cosTheta\right) - cosTheta\right) \cdot \mathsf{PI}\left(\right)}}{cosTheta \cdot \mathsf{PI}\left(\right)} + \left(c + 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (*
    (exp (* (- cosTheta) cosTheta))
    (/ (sqrt (* (- (- 1.0 cosTheta) cosTheta) (PI))) (* cosTheta (PI))))
   (+ c 1.0))))

Derivation

1. Initial program 97.3%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. *-commutative N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. lift-sqrt.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. pow1/2 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{{\color{blue}{\mathsf{PI}\left(\right)}}^{\frac{1}{2}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. add-cube-cbrt N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{\frac{1}{2}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}} \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   10. pow2 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{{\color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)}}^{\frac{1}{2}} \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   11. pow-pow N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(2 \cdot \frac{1}{2}\right)}} \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\color{blue}{1}} \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   13. unpow1 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   14. times-frac N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \frac{1}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \frac{1}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{0.16666666666666666}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\mathsf{PI}\left(\right)}}} \cdot \frac{1}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\frac{1}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. frac-times N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot 1}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. *-rgt-identity N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   7. lift-cbrt.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. pow1/3 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   9. lift-pow.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{{\mathsf{PI}\left(\right)}^{\frac{1}{3}} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   10. pow-prod-up N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{3} + \frac{1}{6}\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{{\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{2}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   12. pow1/2 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   13. lift-sqrt.f32 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   14. associate-/r* N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   16. lower-/.f32 97.9

       \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   17. lift-*.f32 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   18. *-commutative N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   19. lower-*.f32 97.9

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

6. Applied rewrites 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\mathsf{PI}\left(\right)}}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

8. Applied rewrites 98.2%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(\left(1 - cosTheta\right) - cosTheta\right)}}{\mathsf{PI}\left(\right) \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

9. Final simplification 98.2%

   \[\leadsto \frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\sqrt{\left(\left(1 - cosTheta\right) - cosTheta\right) \cdot \mathsf{PI}\left(\right)}}{cosTheta \cdot \mathsf{PI}\left(\right)} + \left(c + 1\right)} \]
10. Add Preprocessing

Alternative 2: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\sqrt{\left(\left(1 - cosTheta\right) - cosTheta\right) \cdot \mathsf{PI}\left(\right)}}{cosTheta \cdot \mathsf{PI}\left(\right)}} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (exp (* (- cosTheta) cosTheta))
    (/ (sqrt (* (- (- 1.0 cosTheta) cosTheta) (PI))) (* cosTheta (PI)))))))

Derivation

1. Initial program 97.3%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   6. associate-*l/ N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{1 \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\mathsf{PI}\left(\right)}}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\mathsf{PI}\left(\right)}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   8. lift-sqrt.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\mathsf{PI}\left(\right)}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   9. lift-sqrt.f32 N/A

      \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   10. sqrt-undiv N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   11. lower-sqrt.f32 N/A

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

   12. lower-/.f32 97.4

       \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt{\color{blue}{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied rewrites 97.4%

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\color{blue}{1} + \frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
6. Step-by-step derivation

7. Applied rewrites 97.1%

   \[\leadsto \frac{1}{\color{blue}{1} + \frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{\sqrt{\frac{\left(1 - cosTheta\right) - cosTheta}{\mathsf{PI}\left(\right)}}}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

9. Applied rewrites 97.9%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(\left(1 - cosTheta\right) - cosTheta\right)}}{\mathsf{PI}\left(\right) \cdot cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

10. Final simplification 97.9%

    \[\leadsto \frac{1}{1 + e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\sqrt{\left(\left(1 - cosTheta\right) - cosTheta\right) \cdot \mathsf{PI}\left(\right)}}{cosTheta \cdot \mathsf{PI}\left(\right)}} \]
11. Add Preprocessing

Alternative 3: 92.9% accurate, 11.4× speedup

Math

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

FPCore

(FPCore (cosTheta c) :precision binary32 (* (sqrt (PI)) cosTheta))

Derivation

1. Initial program 97.3%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing

3. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\mathsf{PI}\left(\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-PI.f32 92.1

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

5. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot cosTheta} \]
6. Add Preprocessing

Alternative 4: 5.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c} \end{array} \]

FPCore

(FPCore (cosTheta c) :precision binary32 (/ 1.0 c))

C

float code(float cosTheta, float c) {
	return 1.0f / c;
}

Fortran

real(4) function code(costheta, c)
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 / c
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / c)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / c;
end

Derivation

1. Initial program 97.3%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{1}{c}} \]
4. Step-by-step derivation

   1. lower-/.f32 5.1

      \[\leadsto \color{blue}{\frac{1}{c}} \]

5. Applied rewrites 5.1%

   \[\leadsto \color{blue}{\frac{1}{c}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt (PI))) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))