Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.8%

Time: 10.8s

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: 97.9% 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: 98.8% 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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. *-rgt-identity N/A

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

   7. times-frac N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. clear-num N/A

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

   11. lift-/.f32 N/A

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

   12. times-frac N/A

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

   13. *-lft-identity N/A

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

   14. lower-/.f32 N/A

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

4. Applied rewrites 98.1%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. div-sub N/A

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

   9. frac-sub N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. sqrt-div N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. sqrt-unprod N/A

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

   16. rem-square-sqrt N/A

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

6. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l/ N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lower-/.f32 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 2: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-PI.f32 N/A

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

   2. add-log-exp N/A

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

   3. *-un-lft-identity N/A

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

   4. lift-PI.f32 N/A

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

   5. exp-prod N/A

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

   6. log-pow N/A

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

   7. lower-*.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. exp-1-e N/A

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

   10. lower-E.f32 98.6

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

9. Applied rewrites 95.7%

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

10. Final simplification 95.7%

    \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} - \left(\left(c + 1\right) - \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \left(cosTheta \cdot \mathsf{PI}\left(\right)\right)\right) \cdot cosTheta \]
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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 93.2

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

5. Applied rewrites 93.2%

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

Alternative 4: 5.1% 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 97.9%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f32 4.9

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

5. Applied rewrites 4.9%

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

Reproduce

herbie shell --seed 2024304 
(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))))))