Beckmann Sample, normalization factor

Percentage Accurate: 97.8% → 98.5%

Time: 4.3s

Alternatives: 14

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: 97.8% 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

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

FPCore

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

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.7724539041519165) / fma(Float32(1.7724539041519165), Float32(c + Float32(1.0)), Float32(sqrt(fma(Float32(-2.0), cosTheta, Float32(1.0))) / Float32(exp(Float32(cosTheta * 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. Step-by-step derivation

3. Applied rewrites 98.4%

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

5. Applied rewrites 98.5%

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

6. Evaluated real constant 98.5%

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

Alternative 2: 98.0% accurate, 1.2× speedup

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

Math

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

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (/
 1.0
 (+
  1.0
  (fma
   (/
    (sqrt (fma -2.0 cosTheta 1.0))
    (* (exp (* cosTheta cosTheta)) cosTheta))
   0.564189612865448
   c))))

C

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

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(sqrt(fma(Float32(-2.0), cosTheta, Float32(1.0))) / Float32(exp(Float32(cosTheta * cosTheta)) * cosTheta)), Float32(0.564189612865448), 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 97.9%

   \[\leadsto \frac{1}{\left(1 + c\right) + \left(0.564189612865448 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
3. Step-by-step derivation

4. Applied rewrites 98.0%

   \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(-2, cosTheta, 1\right)}}{e^{cosTheta \cdot cosTheta} \cdot cosTheta}, 0.564189612865448, c\right)} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 1.3× speedup

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

Math

\[\frac{1}{\left(1 + c\right) + \frac{\frac{1 + cosTheta \cdot \left(cosTheta \cdot \left(0.5 \cdot cosTheta - 1.5\right) - 1\right)}{cosTheta}}{\sqrt{\pi}}} \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (/
 1.0
 (+
  (+ 1.0 c)
  (/
   (/
    (+ 1.0 (* cosTheta (- (* cosTheta (- (* 0.5 cosTheta) 1.5)) 1.0)))
    cosTheta)
   (sqrt PI)))))

C

float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f + (cosTheta * ((cosTheta * ((0.5f * cosTheta) - 1.5f)) - 1.0f))) / cosTheta) / sqrtf(((float) M_PI))));
}

Julia

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) + Float32(cosTheta * Float32(Float32(cosTheta * Float32(Float32(Float32(0.5) * cosTheta) - Float32(1.5))) - Float32(1.0)))) / cosTheta) / sqrt(Float32(pi)))))
end

MATLAB

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) + (cosTheta * ((cosTheta * ((single(0.5) * cosTheta) - single(1.5))) - single(1.0)))) / cosTheta) / sqrt(single(pi))));
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. Step-by-step derivation

3. Applied rewrites 98.4%

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

4. Taylor expanded in cosTheta around 0

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

6. Applied rewrites 97.4%

   \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\frac{1 + cosTheta \cdot \left(cosTheta \cdot \left(0.5 \cdot cosTheta - 1.5\right) - 1\right)}{cosTheta}}{\sqrt{\pi}}} \]
7. Add Preprocessing

Alternative 4: 97.1% accurate, 1.8× speedup

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

Math

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

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
    (fma 0.282094806432724 cosTheta -0.846284419298172)
    cosTheta
    (+ 0.435810387134552 c))
   cosTheta
   0.564189612865448)
  cosTheta)))

C

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

Julia

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

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

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 97.0%

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

7. Applied rewrites 97.0%

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

Alternative 5: 97.0% accurate, 2.0× speedup

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

Math

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

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
   (+ 0.435810387134552 c))
  (/ 0.564189612865448 cosTheta))))

C

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

Julia

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

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

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 97.0%

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

7. Applied rewrites 97.1%

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

Alternative 6: 96.8% accurate, 2.3× speedup

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

Math

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

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
   0.435810387134552)
  (/ 0.564189612865448 cosTheta))))

C

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

Julia

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

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

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 97.0%

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

6. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\frac{0.564189612865448 + cosTheta \cdot \left(\frac{7311685}{16777216} + cosTheta \cdot \left(\frac{9465531}{33554432} \cdot cosTheta - \frac{28396593}{33554432}\right)\right)}{cosTheta}} \]
7. Step-by-step derivation

8. Applied rewrites 96.8%

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

10. Applied rewrites 96.8%

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

Alternative 7: 96.4% accurate, 2.5× speedup

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

Math

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

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
  -0.846284419298172
  cosTheta
  (+ (+ 0.435810387134552 c) (/ 0.564189612865448 cosTheta)))))

C

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

Julia

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

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 96.4%

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

7. Applied rewrites 96.4%

   \[\leadsto \frac{1}{\mathsf{fma}\left(-0.846284419298172, cosTheta, 0.435810387134552 + c\right) + \frac{0.564189612865448}{cosTheta}} \]
8. Step-by-step derivation

9. Applied rewrites 96.4%

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

Alternative 8: 96.4% accurate, 2.5× speedup

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

Math

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

FPCore

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

C

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

Julia

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

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 96.3%

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

10. Applied rewrites 96.4%

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

Alternative 9: 96.4% accurate, 2.5× speedup

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

Math

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

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 -0.846284419298172 cosTheta (+ 0.435810387134552 c))
  (/ 0.564189612865448 cosTheta))))

C

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

Julia

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

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 96.4%

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

7. Applied rewrites 96.4%

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

Alternative 10: 96.3% accurate, 3.4× speedup

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

Math

\[\mathsf{fma}\left(\mathsf{fma}\left(3.716276871595387, cosTheta, -1.3691385682874957\right), cosTheta, 1.7724537588012759\right) \cdot cosTheta \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (*
 (fma
  (fma 3.716276871595387 cosTheta -1.3691385682874957)
  cosTheta
  1.7724537588012759)
 cosTheta))

C

float code(float cosTheta, float c) {
	return fmaf(fmaf(3.716276871595387f, cosTheta, -1.3691385682874957f), cosTheta, 1.7724537588012759f) * cosTheta;
}

Julia

function code(cosTheta, c)
	return Float32(fma(fma(Float32(3.716276871595387), cosTheta, Float32(-1.3691385682874957)), cosTheta, Float32(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%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto cosTheta \cdot \left(\frac{16777216}{9465531} + cosTheta \cdot \left(cosTheta \cdot \left(\frac{8388608}{3155177} - \frac{-4722366482869645213696}{848076338487857316291} \cdot {\left(\frac{7311685}{16777216} + c\right)}^{2}\right) - \frac{281474976710656}{89596277111961} \cdot \left(\frac{7311685}{16777216} + c\right)\right)\right) \]
4. Step-by-step derivation

5. Applied rewrites 96.4%

   \[\leadsto cosTheta \cdot \left(1.7724537588012759 + cosTheta \cdot \left(cosTheta \cdot \left(2.658680638201914 - -5.568327128769741 \cdot {\left(0.435810387134552 + c\right)}^{2}\right) - 3.141592327088772 \cdot \left(0.435810387134552 + c\right)\right)\right) \]

6. Taylor expanded in c around 0

   \[\leadsto cosTheta \cdot \left(1.7724537588012759 + cosTheta \cdot \left(\frac{3151686482069725118464}{848076338487857316291} \cdot cosTheta - \frac{122669718568960}{89596277111961}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.3%

   \[\leadsto cosTheta \cdot \left(1.7724537588012759 + cosTheta \cdot \left(3.716276871595387 \cdot cosTheta - 1.3691385682874957\right)\right) \]

10. Applied rewrites 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(3.716276871595387, cosTheta, -1.3691385682874957\right), cosTheta, 1.7724537588012759\right) \cdot cosTheta \]
11. Add Preprocessing

Alternative 11: 95.2% accurate, 4.0× speedup

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

Math

\[\mathsf{fma}\left(cosTheta, -1.3691385682874957 \cdot cosTheta, 1.7724537588012759 \cdot cosTheta\right) \]

FPCore

(FPCore (cosTheta c)
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999))
     (and (< -1.0 c) (< c 1.0)))
  (fma
 cosTheta
 (* -1.3691385682874957 cosTheta)
 (* 1.7724537588012759 cosTheta)))

C

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

Julia

function code(cosTheta, c)
	return fma(cosTheta, Float32(Float32(-1.3691385682874957) * cosTheta), Float32(Float32(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%

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 95.3%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 95.1%

   \[\leadsto cosTheta \cdot \left(1.7724537588012759 + -1.3691385682874957 \cdot cosTheta\right) \]
9. Step-by-step derivation

10. Applied rewrites 95.2%

    \[\leadsto \mathsf{fma}\left(cosTheta, -1.3691385682874957 \cdot cosTheta, 1.7724537588012759 \cdot cosTheta\right) \]
11. Add Preprocessing

Alternative 12: 95.1% accurate, 5.0× speedup

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

Math

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

FPCore

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

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 95.3%

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

6. Taylor expanded in c around 0

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

8. Applied rewrites 95.1%

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

Alternative 13: 92.7% 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 97.8%

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

2. Taylor expanded in cosTheta around 0

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

4. Applied rewrites 92.7%

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

5. Evaluated real constant 92.7%

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

Alternative 14: 92.5% 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 97.8%

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

2. Evaluated real constant 97.9%

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 92.5%

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

Reproduce

herbie shell --seed 2026084 
(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))))))