Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 8.7s

Alternatives: 8

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, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (* cosTheta (sqrt (PI)))) (sqrt (- (- 1.0 cosTheta) 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. 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. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r/ N/A

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

   8. *-rgt-identity N/A

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

   9. times-frac N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f32 N/A

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

4. Applied rewrites 98.8%

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

Alternative 2: 98.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) (* cosTheta (sqrt (PI))))
    (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. lift-/.f32 N/A

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

   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 98.8

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

4. Applied rewrites 98.8%

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

Alternative 3: 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

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

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. Add Preprocessing

Alternative 4: 95.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (* cosTheta (sqrt (PI)))) (- 1.0 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. 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. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r/ N/A

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

   8. *-rgt-identity N/A

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

   9. times-frac N/A

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

   10. un-div-inv N/A

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

   11. lift-/.f32 N/A

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

4. Applied rewrites 98.8%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. lift--.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. associate-+l- N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 98.7

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 97.1

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

9. Applied rewrites 97.1%

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

Alternative 5: 95.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{1}{\left(1 + c\right) + \frac{1}{\frac{\left(\left(cosTheta \cdot cosTheta\right) \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right) \cdot cosTheta}{t\_0 \cdot cosTheta - t\_0}}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (sqrt (PI))))
   (/
    1.0
    (+
     (+ 1.0 c)
     (/
      1.0
      (/
       (* (- (* (* cosTheta cosTheta) (PI)) (PI)) cosTheta)
       (- (* t_0 cosTheta) t_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. 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

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-PI.f32 94.2

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

7. Applied rewrites 93.9%

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

9. Applied rewrites 96.8%

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

Alternative 6: 95.3% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/ 1.0 (+ (+ 1.0 c) (/ 1.0 (* (* (+ cosTheta 1.0) (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. 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

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

   3. *-commutative N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l/ N/A

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

   7. clear-num N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-PI.f32 94.2

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

7. Applied rewrites 93.9%

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

9. Applied rewrites 96.6%

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

Alternative 7: 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 94.4

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

5. Applied rewrites 94.4%

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

Alternative 8: 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 2024318 
(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))))))