Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 10.3s

Alternatives: 10

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: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lift-sqrt.f32 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-/.f32 97.6

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

7. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. inv-pow N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower-/.f32 98.2

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

9. Applied rewrites 98.2%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. lower-+.f32 98.7

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

12. Applied rewrites 98.7%

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

13. Final simplification 98.7%

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

Alternative 2: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 98.6

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

4. Applied rewrites 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lift-sqrt.f32 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-/.f32 97.6

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

7. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. inv-pow N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower-/.f32 98.2

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 98.2

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

4. Applied rewrites 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-/.f32 97.8

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

7. Applied rewrites 97.8%

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

8. Final simplification 97.8%

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

Alternative 6: 96.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. mul-1-neg N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. *-lft-identity N/A

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

   6. distribute-rgt-out-- N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower--.f32 96.7

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

5. Applied rewrites 96.7%

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

7. Applied rewrites 97.2%

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

9. Applied rewrites 97.2%

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

10. Final simplification 97.2%

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

Alternative 7: 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 97.8%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l/ N/A

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

   7. *-lft-identity N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. sqrt-undiv N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-/.f32 97.8

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

7. Applied rewrites 97.8%

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

8. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 97.0%

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

11. Final simplification 97.0%

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

Alternative 8: 94.9% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 97.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lift-sqrt.f32 N/A

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

   12. sqrt-undiv N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-/.f32 97.6

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

7. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. inv-pow N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower-/.f32 98.2

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

9. Applied rewrites 98.2%

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

10. Taylor expanded in cosTheta around 0

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

    1. associate-*r* N/A

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

    2. lower-/.f32 N/A

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

    3. distribute-rgt1-in N/A

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

    4. +-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. mul-1-neg N/A

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

    7. unsub-neg N/A

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

    8. lower--.f32 N/A

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

    9. lower-sqrt.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. lower-PI.f32 96.2

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

12. Applied rewrites 96.2%

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

13. Final simplification 96.2%

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

Alternative 9: 92.8% 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 97.8%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 94.0

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

5. Applied rewrites 94.0%

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

Alternative 10: 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 97.8%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f32 5.0

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

5. Applied rewrites 5.0%

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

Reproduce

herbie shell --seed 2024271 
(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))))))