Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.0%

Time: 17.3s

Alternatives: 7

Speedup: 1.1×

• unsoundness detected in the egraph

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
  (/
 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: 97.8% accurate, 1.0× speedup

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
  (/
 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: 98.0% accurate, 1.1× speedup

Math

\[\frac{1}{\left(1 + c\right) + \left(0.564189612865448 \cdot \frac{\sqrt{1 + -2 \cdot cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (/
 1.0
 (+
  (+ 1.0 c)
  (*
   (* 0.564189612865448 (/ (sqrt (+ 1.0 (* -2.0 cosTheta))) cosTheta))
   (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((0.564189612865448f * (sqrtf((1.0f + (-2.0f * cosTheta))) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 / ((1.0e0 + c) + ((0.564189612865448e0 * (sqrt((1.0e0 + ((-2.0e0) * costheta))) / costheta)) * exp((-costheta * costheta))))
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(0.564189612865448) * Float32(sqrt(Float32(Float32(1.0) + Float32(Float32(-2.0) * cosTheta))) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((single(0.564189612865448) * (sqrt((single(1.0) + (single(-2.0) * cosTheta))) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in cosTheta around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 97.8%

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

4. Applied rewrites 97.8%

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

5. Evaluated real constant 98.0%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{0.564189612865448} \cdot \frac{\sqrt{1 + -2 \cdot cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
6. Add Preprocessing

Alternative 2: 97.8% accurate, 1.1× speedup

Math

\[\frac{1}{\left(1 + c\right) + \left(0.5641895532608032 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (/
 1.0
 (+
  (+ 1.0 c)
  (*
   (*
    0.5641895532608032
    (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
   (exp (* (- cosTheta) cosTheta))))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + ((0.5641895532608032f * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = 1.0e0 / ((1.0e0 + c) + ((0.5641895532608032e0 * (sqrt(((1.0e0 - costheta) - costheta)) / costheta)) * exp((-costheta * costheta))))
end function

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(0.5641895532608032) * 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(0.5641895532608032) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end

Derivation

1. Initial program 97.8%

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

2. Evaluated real constant 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\color{blue}{1.7724539041519165}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

3. Evaluated real constant 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\color{blue}{0.5641895532608032} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
4. Add Preprocessing

Alternative 3: 95.7% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (*
 cosTheta
 (+
  1.7724539041519165
  (* -3.141592842343371 (* cosTheta (+ 0.4358104334010989 c))))))

C

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

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 + ((-3.141592842343371e0) * (costheta * (0.4358104334010989e0 + c))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (single(1.7724539041519165) + (single(-3.141592842343371) * (cosTheta * (single(0.4358104334010989) + c))));
end

Derivation

1. Initial program 97.8%

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

2. Evaluated real constant 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\color{blue}{1.7724539041519165}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \color{blue}{\frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \color{blue}{\left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \color{blue}{\left(\frac{6479813}{14868421} + c\right)}\right)\right) \]

   5. lower-+.f32 95.7%

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

5. Applied rewrites 95.7%

   \[\leadsto \color{blue}{cosTheta \cdot \left(1.7724539041519165 + -3.141592842343371 \cdot \left(cosTheta \cdot \left(0.4358104334010989 + c\right)\right)\right)} \]
6. Add Preprocessing

Alternative 4: 95.5% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (* 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 97.8%

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

2. Evaluated real constant 97.8%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{1}{\color{blue}{1.7724539041519165}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \color{blue}{\frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \color{blue}{\left(cosTheta \cdot \left(\frac{6479813}{14868421} + c\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{14868421}{8388608} + \frac{-221069943033241}{70368744177664} \cdot \left(cosTheta \cdot \color{blue}{\left(\frac{6479813}{14868421} + c\right)}\right)\right) \]

   5. lower-+.f32 95.7%

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in c around 0

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

   1. lower-*.f32 95.5%

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

8. Applied rewrites 95.5%

   \[\leadsto cosTheta \cdot \left(1.7724539041519165 + -1.3691389381914547 \cdot \color{blue}{cosTheta}\right) \]
9. Add Preprocessing

Alternative 5: 95.3% accurate, 5.0× speedup

Math

\[cosTheta \cdot \left(1.7724537588012759 + -1.3691385682874957 \cdot cosTheta\right) \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (* cosTheta (+ 1.7724537588012759 (* -1.3691385682874957 cosTheta))))

C

float code(float cosTheta, float c) {
	return cosTheta * (1.7724537588012759f + (-1.3691385682874957f * cosTheta));
}

Fortran

real(4) function code(costheta, c)
use fmin_fmax_functions
    real(4), intent (in) :: costheta
    real(4), intent (in) :: c
    code = costheta * (1.7724537588012759e0 + ((-1.3691385682874957e0) * costheta))
end function

Julia

function code(cosTheta, c)
	return Float32(cosTheta * Float32(Float32(1.7724537588012759) + Float32(Float32(-1.3691385682874957) * cosTheta)))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = cosTheta * (single(1.7724537588012759) + (single(-1.3691385682874957) * cosTheta));
end

Derivation

1. Initial program 97.8%

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

2. Evaluated real constant 97.9%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{cosTheta \cdot \left(\frac{16777216}{9465531} + \frac{-281474976710656}{89596277111961} \cdot \left(cosTheta \cdot \left(\frac{7311685}{16777216} + c\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \color{blue}{\left(\frac{16777216}{9465531} + \frac{-281474976710656}{89596277111961} \cdot \left(cosTheta \cdot \left(\frac{7311685}{16777216} + c\right)\right)\right)} \]

   2. lower-+.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{16777216}{9465531} + \color{blue}{\frac{-281474976710656}{89596277111961} \cdot \left(cosTheta \cdot \left(\frac{7311685}{16777216} + c\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{16777216}{9465531} + \frac{-281474976710656}{89596277111961} \cdot \color{blue}{\left(cosTheta \cdot \left(\frac{7311685}{16777216} + c\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto cosTheta \cdot \left(\frac{16777216}{9465531} + \frac{-281474976710656}{89596277111961} \cdot \left(cosTheta \cdot \color{blue}{\left(\frac{7311685}{16777216} + c\right)}\right)\right) \]

   5. lower-+.f32 95.4%

      \[\leadsto cosTheta \cdot \left(1.7724537588012759 + -3.141592327088772 \cdot \left(cosTheta \cdot \left(0.435810387134552 + \color{blue}{c}\right)\right)\right) \]

5. Applied rewrites 95.4%

   \[\leadsto \color{blue}{cosTheta \cdot \left(1.7724537588012759 + -3.141592327088772 \cdot \left(cosTheta \cdot \left(0.435810387134552 + c\right)\right)\right)} \]

6. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(1.7724537588012759 + \frac{-122669718568960}{89596277111961} \cdot \color{blue}{cosTheta}\right) \]
7. Step-by-step derivation

   1. lower-*.f32 95.3%

      \[\leadsto cosTheta \cdot \left(1.7724537588012759 + -1.3691385682874957 \cdot cosTheta\right) \]

8. Applied rewrites 95.3%

   \[\leadsto cosTheta \cdot \left(1.7724537588012759 + -1.3691385682874957 \cdot \color{blue}{cosTheta}\right) \]
9. Add Preprocessing

Alternative 6: 92.8% accurate, 11.9× speedup

Math

\[cosTheta \cdot 1.7724539041519165 \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (* 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 97.8%

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

2. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-PI.f32 92.8%

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

4. Applied rewrites 92.8%

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

5. Evaluated real constant 92.8%

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

Alternative 7: 92.7% accurate, 11.9× speedup

Math

\[1.7724537588012759 \cdot cosTheta \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  (* 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 97.8%

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

2. Evaluated real constant 97.9%

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

3. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 92.7%

      \[\leadsto 1.7724537588012759 \cdot \color{blue}{cosTheta} \]

5. Applied rewrites 92.7%

   \[\leadsto \color{blue}{1.7724537588012759 \cdot cosTheta} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(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))))))