Beckmann Sample, normalization factor

Percentage Accurate: 97.9% → 98.5%

Time: 4.0s

Alternatives: 9

Speedup: 1.2×

Specification

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

Math

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

FPCore

(FPCore (cosTheta c)
  :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))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Initial Program: 97.9% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (cosTheta c)
  :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))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Alternative 1: 98.5% accurate, 1.2× speedup

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

Math

\[\frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot 1.7724539041519165}, c + 1\right)} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (/
 1.0
 (fma
  (sqrt (fma -2.0 cosTheta 1.0))
  (/ (exp (* (- cosTheta) cosTheta)) (* cosTheta 1.7724539041519165))
  (+ c 1.0))))

C

float code(float cosTheta, float c) {
	return 1.0f / fmaf(sqrtf(fmaf(-2.0f, cosTheta, 1.0f)), (expf((-cosTheta * cosTheta)) / (cosTheta * 1.7724539041519165f)), (c + 1.0f));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / fma(sqrt(fma(Float32(-2.0), cosTheta, Float32(1.0))), Float32(exp(Float32(Float32(-cosTheta) * cosTheta)) / Float32(cosTheta * Float32(1.7724539041519165))), Float32(c + Float32(1.0))))
end

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

3. Applied rewrites 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot \sqrt{\pi}}, c + 1\right)} \]

4. Evaluated real constant 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot 1.7724539041519165}, c + 1\right)} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.5× speedup

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

Math

\[\frac{1}{\left(\left(c + 1\right) + \frac{-1.5 \cdot cosTheta - 1}{\sqrt{\pi}}\right) + \frac{1}{cosTheta \cdot \sqrt{\pi}}} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (/
 1.0
 (+
  (+ (+ c 1.0) (/ (- (* -1.5 cosTheta) 1.0) (sqrt PI)))
  (/ 1.0 (* cosTheta (sqrt PI))))))

C

float code(float cosTheta, float c) {
	return 1.0f / (((c + 1.0f) + (((-1.5f * cosTheta) - 1.0f) / sqrtf(((float) M_PI)))) + (1.0f / (cosTheta * sqrtf(((float) M_PI)))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(c + Float32(1.0)) + Float32(Float32(Float32(Float32(-1.5) * cosTheta) - Float32(1.0)) / sqrt(Float32(pi)))) + Float32(Float32(1.0) / Float32(cosTheta * sqrt(Float32(pi))))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / (((c + single(1.0)) + (((single(-1.5) * cosTheta) - single(1.0)) / sqrt(single(pi)))) + (single(1.0) / (cosTheta * sqrt(single(pi)))));
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{\frac{cosTheta \cdot \left(\left(1 + \left(c + \frac{-3}{2} \cdot \frac{cosTheta}{\sqrt{\pi}}\right)\right) - \frac{1}{\sqrt{\pi}}\right) + \frac{1}{\sqrt{\pi}}}{cosTheta}} \]
3. Step-by-step derivation

4. Applied rewrites 96.4%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(cosTheta, \left(1 + \left(c + -1.5 \cdot \frac{cosTheta}{\sqrt{\pi}}\right)\right) - \frac{1}{\sqrt{\pi}}, \frac{1}{\sqrt{\pi}}\right)}{cosTheta}} \]

6. Applied rewrites 97.0%

   \[\leadsto \frac{1}{\left(\left(c + 1\right) + \frac{-1.5 \cdot cosTheta - 1}{\sqrt{\pi}}\right) + \frac{1}{cosTheta \cdot \sqrt{\pi}}} \]
7. Add Preprocessing

Alternative 3: 97.1% accurate, 1.8× speedup

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

Math

\[\frac{1}{\frac{0.564189612865448 + \left(\mathsf{fma}\left(\mathsf{fma}\left(0.282094806432724, cosTheta, -0.846284419298172\right), cosTheta, c\right) + 0.435810387134552\right) \cdot cosTheta}{cosTheta}} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (/
 1.0
 (/
  (+
   0.564189612865448
   (*
    (+
     (fma
      (fma 0.282094806432724 cosTheta -0.846284419298172)
      cosTheta
      c)
     0.435810387134552)
    cosTheta))
  cosTheta)))

C

float code(float cosTheta, float c) {
	return 1.0f / ((0.564189612865448f + ((fmaf(fmaf(0.282094806432724f, cosTheta, -0.846284419298172f), cosTheta, c) + 0.435810387134552f) * cosTheta)) / cosTheta);
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(0.564189612865448) + Float32(Float32(fma(fma(Float32(0.282094806432724), cosTheta, Float32(-0.846284419298172)), cosTheta, c) + Float32(0.435810387134552)) * cosTheta)) / cosTheta))
end

Derivation

1. Initial program 97.9%

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

2. Evaluated real constant 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(0.564189612865448 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{\frac{\frac{9465531}{16777216} + cosTheta \cdot \left(\frac{7311685}{16777216} + \left(c + cosTheta \cdot \left(\frac{9465531}{33554432} \cdot cosTheta - \frac{28396593}{33554432}\right)\right)\right)}{cosTheta}} \]
4. Step-by-step derivation

5. Applied rewrites 97.1%

   \[\leadsto \frac{1}{\frac{0.564189612865448 + cosTheta \cdot \left(0.435810387134552 + \left(c + cosTheta \cdot \left(0.282094806432724 \cdot cosTheta - 0.846284419298172\right)\right)\right)}{cosTheta}} \]
6. Step-by-step derivation

7. Applied rewrites 97.1%

   \[\leadsto \frac{1}{\frac{0.564189612865448 + \left(\mathsf{fma}\left(\mathsf{fma}\left(0.282094806432724, cosTheta, -0.846284419298172\right), cosTheta, c\right) + 0.435810387134552\right) \cdot cosTheta}{cosTheta}} \]
8. Add Preprocessing

Alternative 4: 97.0% accurate, 2.0× speedup

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

Math

\[\frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.282094806432724, cosTheta, -0.846284419298172\right), cosTheta, c\right) + 0.435810387134552\right) + \frac{0.564189612865448}{cosTheta}} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (/
 1.0
 (+
  (+
   (fma
    (fma 0.282094806432724 cosTheta -0.846284419298172)
    cosTheta
    c)
   0.435810387134552)
  (/ 0.564189612865448 cosTheta))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((fmaf(fmaf(0.282094806432724f, cosTheta, -0.846284419298172f), cosTheta, c) + 0.435810387134552f) + (0.564189612865448f / cosTheta));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(fma(fma(Float32(0.282094806432724), cosTheta, Float32(-0.846284419298172)), cosTheta, c) + Float32(0.435810387134552)) + Float32(Float32(0.564189612865448) / cosTheta)))
end

Derivation

1. Initial program 97.9%

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

2. Evaluated real constant 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(0.564189612865448 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{1}{\frac{\frac{9465531}{16777216} + cosTheta \cdot \left(\frac{7311685}{16777216} + \left(c + cosTheta \cdot \left(\frac{9465531}{33554432} \cdot cosTheta - \frac{28396593}{33554432}\right)\right)\right)}{cosTheta}} \]
4. Step-by-step derivation

5. Applied rewrites 97.1%

   \[\leadsto \frac{1}{\frac{0.564189612865448 + cosTheta \cdot \left(0.435810387134552 + \left(c + cosTheta \cdot \left(0.282094806432724 \cdot cosTheta - 0.846284419298172\right)\right)\right)}{cosTheta}} \]

7. Applied rewrites 97.1%

   \[\leadsto \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.282094806432724, cosTheta, -0.846284419298172\right), cosTheta, c\right) + 0.435810387134552\right) + \frac{0.564189612865448}{cosTheta}} \]
8. Add Preprocessing

Alternative 5: 96.7% accurate, 3.1× speedup

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

Math

\[cosTheta \cdot \left(1.7724539041519165 + cosTheta \cdot \left(3.7162775743602734 \cdot cosTheta - 1.3691389381914547\right)\right) \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (*
 cosTheta
 (+
  1.7724539041519165
  (*
   cosTheta
   (- (* 3.7162775743602734 cosTheta) 1.3691389381914547)))))

C

float code(float cosTheta, float c) {
	return cosTheta * (1.7724539041519165f + (cosTheta * ((3.7162775743602734f * cosTheta) - 1.3691389381914547f)));
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = costheta * (1.7724539041519165e0 + (costheta * ((3.7162775743602734e0 * costheta) - 1.3691389381914547e0)))
end function

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(Float32(1.7724539041519165) + Float32(cosTheta * Float32(Float32(Float32(3.7162775743602734) * cosTheta) - Float32(1.3691389381914547)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (single(1.7724539041519165) + (cosTheta * ((single(3.7162775743602734) * cosTheta) - single(1.3691389381914547))));
end

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

3. Applied rewrites 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot \sqrt{\pi}}, c + 1\right)} \]

4. Evaluated real constant 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot 1.7724539041519165}, c + 1\right)} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + cosTheta \cdot \left(cosTheta \cdot \left(\frac{44605263}{16777216} - \frac{-3286960983464244182461}{590295810358705651712} \cdot {\left(\frac{6479813}{14868421} + c\right)}^{2}\right) - \frac{221069943033241}{70368744177664} \cdot \left(\frac{6479813}{14868421} + c\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.9%

   \[\leadsto cosTheta \cdot \left(1.7724539041519165 + cosTheta \cdot \left(cosTheta \cdot \left(2.6586808562278748 - -5.568328498667225 \cdot {\left(0.4358104334010989 + c\right)}^{2}\right) - 3.141592842343371 \cdot \left(0.4358104334010989 + c\right)\right)\right) \]

8. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(1.7724539041519165 + cosTheta \cdot \left(\frac{2193703082274882616765}{590295810358705651712} \cdot cosTheta - \frac{96344587685273}{70368744177664}\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.7%

    \[\leadsto cosTheta \cdot \left(1.7724539041519165 + cosTheta \cdot \left(3.7162775743602734 \cdot cosTheta - 1.3691389381914547\right)\right) \]
11. Add Preprocessing

Alternative 6: 95.8% accurate, 3.3× speedup

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

Math

\[\mathsf{fma}\left(-3.141592842343371, \left(0.4358104334010989 + c\right) \cdot cosTheta, 1.7724539041519165\right) \cdot cosTheta \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (*
 (fma
  -3.141592842343371
  (* (+ 0.4358104334010989 c) cosTheta)
  1.7724539041519165)
 cosTheta))

C

float code(float cosTheta, float c) {
	return fmaf(-3.141592842343371f, ((0.4358104334010989f + c) * cosTheta), 1.7724539041519165f) * cosTheta;
}

Julia

function code(cosTheta, c)
	return Float32(fma(Float32(-3.141592842343371), Float32(Float32(Float32(0.4358104334010989) + c) * cosTheta), Float32(1.7724539041519165)) * cosTheta)
end

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

3. Applied rewrites 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot \sqrt{\pi}}, c + 1\right)} \]

4. Evaluated real constant 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot 1.7724539041519165}, c + 1\right)} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

   \[\leadsto cosTheta \cdot \left(1.7724539041519165 + -3.141592842343371 \cdot \left(cosTheta \cdot \left(0.4358104334010989 + c\right)\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 95.8%

   \[\leadsto \mathsf{fma}\left(-3.141592842343371, \left(0.4358104334010989 + c\right) \cdot cosTheta, 1.7724539041519165\right) \cdot cosTheta \]
10. Add Preprocessing

Alternative 7: 95.7% accurate, 5.0× speedup

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

Math

\[cosTheta \cdot \left(1.7724539041519165 + -1.3691389381914547 \cdot cosTheta\right) \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (* cosTheta (+ 1.7724539041519165 (* -1.3691389381914547 cosTheta))))

C

float code(float cosTheta, float c) {
	return cosTheta * (1.7724539041519165f + (-1.3691389381914547f * cosTheta));
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = costheta * (1.7724539041519165e0 + ((-1.3691389381914547e0) * costheta))
end function

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(Float32(1.7724539041519165) + Float32(Float32(-1.3691389381914547) * cosTheta)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (single(1.7724539041519165) + (single(-1.3691389381914547) * cosTheta));
end

Derivation

1. Initial program 97.9%

   \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
2. Step-by-step derivation

3. Applied rewrites 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot \sqrt{\pi}}, c + 1\right)} \]

4. Evaluated real constant 98.5%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}, \frac{e^{\left(-cosTheta\right) \cdot cosTheta}}{cosTheta \cdot 1.7724539041519165}, c + 1\right)} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

   \[\leadsto cosTheta \cdot \left(1.7724539041519165 + -3.141592842343371 \cdot \left(cosTheta \cdot \left(0.4358104334010989 + c\right)\right)\right) \]

8. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(1.7724539041519165 + \frac{-96344587685273}{70368744177664} \cdot cosTheta\right) \]
9. Step-by-step derivation

10. Applied rewrites 95.7%

    \[\leadsto cosTheta \cdot \left(1.7724539041519165 + -1.3691389381914547 \cdot cosTheta\right) \]
11. Add Preprocessing

Alternative 8: 93.0% accurate, 11.9× speedup

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

Math

\[cosTheta \cdot 1.7724539041519165 \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (* cosTheta 1.7724539041519165))

C

float code(float cosTheta, float c) {
	return cosTheta * 1.7724539041519165f;
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = costheta * 1.7724539041519165e0
end function

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(1.7724539041519165))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * single(1.7724539041519165);
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in cosTheta around 0

   \[\leadsto cosTheta \cdot \sqrt{\pi} \]
3. Step-by-step derivation

4. Applied rewrites 93.0%

   \[\leadsto cosTheta \cdot \sqrt{\pi} \]

5. Evaluated real constant 93.0%

   \[\leadsto cosTheta \cdot 1.7724539041519165 \]
6. Add Preprocessing

Alternative 9: 92.9% accurate, 11.9× speedup

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

Math

\[1.7724537588012759 \cdot cosTheta \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (* 1.7724537588012759 cosTheta))

C

float code(float cosTheta, float c) {
	return 1.7724537588012759f * cosTheta;
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.7724537588012759e0 * costheta
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.7724537588012759) * cosTheta)
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.7724537588012759) * cosTheta;
end

Derivation

1. Initial program 97.9%

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

2. Evaluated real constant 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(0.564189612865448 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{16777216}{9465531} \cdot cosTheta \]
4. Step-by-step derivation

5. Applied rewrites 92.9%

   \[\leadsto 1.7724537588012759 \cdot cosTheta \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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))))))