Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.6%

Time: 10.2s

Alternatives: 12

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: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\frac{\sqrt{\left(1 - 2 \cdot cosTheta\right) \cdot \mathsf{PI}\left(\right)}}{cosTheta}}{\mathsf{PI}\left(\right)} + \left(c + 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (*
    (exp (* (- cosTheta) cosTheta))
    (/ (/ (sqrt (* (- 1.0 (* 2.0 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

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

   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 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. div-sub N/A

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

   5. frac-sub N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. sqrt-div N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. sqrt-unprod N/A

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

   12. rem-square-sqrt N/A

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

   13. lower-/.f32 N/A

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

6. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

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

   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 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. div-sub N/A

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

   5. frac-sub N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. sqrt-div N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. sqrt-unprod N/A

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

   12. rem-square-sqrt N/A

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

   13. lower-/.f32 N/A

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

6. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

8. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. count-2 N/A

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

   5. associate--l- N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 98.8

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

10. Applied rewrites 98.8%

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

11. Final simplification 98.8%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\sqrt{\left(1 - 2 \cdot cosTheta\right) \cdot \mathsf{PI}\left(\right)}}{cosTheta \cdot \mathsf{PI}\left(\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + \left(c + 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (*
    (/ (sqrt (* (- 1.0 (* 2.0 cosTheta)) (PI))) (* cosTheta (PI)))
    (exp (* (- cosTheta) cosTheta)))
   (+ 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

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

   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 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. div-sub N/A

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

   5. frac-sub N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. sqrt-div N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. sqrt-unprod N/A

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

   12. rem-square-sqrt N/A

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

   13. lower-/.f32 N/A

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

6. Applied rewrites 98.2%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. 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%

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

9. Final simplification 98.8%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (*
    (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) (* (sqrt (PI)) cosTheta))
    (exp (* (- cosTheta) cosTheta)))
   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. 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 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in c around 0

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

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

   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 98.0

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

4. Applied rewrites 98.0%

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

5. Final simplification 98.0%

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

Alternative 6: 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 97.9%

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

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

   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 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in c around 0

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

7. Applied rewrites 97.7%

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

8. Final simplification 97.7%

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

Alternative 7: 95.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (*
    (/ (- 1.0 cosTheta) (* (sqrt (PI)) cosTheta))
    (exp (* (- cosTheta) cosTheta)))
   (+ 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. 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 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in cosTheta around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 95.9

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

7. Applied rewrites 95.9%

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

8. Final simplification 95.9%

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

Alternative 8: 95.8% accurate, 2.8× speedup

Math

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

FPCore

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

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

   2. lift-+.f32 N/A

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

   3. flip3-+ N/A

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

   4. clear-num N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*r/ N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 91.3

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

7. Applied rewrites 91.3%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f32 N/A

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

   17. lower-sqrt.f32 N/A

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

10. Applied rewrites 94.8%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-/r* N/A

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

12. Applied rewrites 95.6%

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

Alternative 9: 95.1% accurate, 3.0× speedup

Math

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

FPCore

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

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

   2. lift-+.f32 N/A

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

   3. flip3-+ N/A

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

   4. clear-num N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*r/ N/A

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

   10. associate-*r/ N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

   3. lower-/.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 91.3

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

7. Applied rewrites 91.3%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f32 N/A

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

   17. lower-sqrt.f32 N/A

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

10. Applied rewrites 94.8%

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

12. Applied rewrites 94.9%

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

Alternative 10: 95.0% accurate, 3.2× speedup

Math

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

FPCore

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

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

   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 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. associate-*r* N/A

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

   2. lower-/.f32 N/A

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

   3. distribute-rgt1-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 94.7

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

7. Applied rewrites 94.7%

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

8. Final simplification 94.7%

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

Alternative 11: 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 92.4

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

5. Applied rewrites 92.4%

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

Alternative 12: 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 2024255 
(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))))))