Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 8.6s

Alternatives: 5

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.5% accurate, 1.1× speedup

Math

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

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 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
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 98.8

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

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 2: 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 97.9%

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

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

4. Applied rewrites 98.3%

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

Alternative 3: 96.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-/.f32 N/A

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

   4. mul-1-neg N/A

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

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

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-out-- N/A

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

   8. lower-*.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower--.f32 96.2

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

5. Applied rewrites 96.2%

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

7. Applied rewrites 96.3%

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

9. Applied rewrites 96.8%

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

Alternative 4: 93.1% accurate, 11.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-PI.f32 93.6

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

5. Applied rewrites 93.6%

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

Alternative 5: 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 97.9%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f32 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 2024313 
(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))))))