Beckmann Sample, normalization factor

Average Error: 2.13% → 1.47%

Time: 14.9s

Precision: binary32

Cost: 22848

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

Math

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

↓

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

FPCore

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

↓

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

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))));
}

↓

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

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

↓

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

Derivation

1. Initial program 2.13

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

2. Simplified 1.47

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

   [Start] 2.13	\[ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   +-commutative [=>] 2.13	\[ \frac{1}{\color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta} + \left(1 + c\right)}} \]
   *-commutative [=>] 2.13	\[ \frac{1}{\color{blue}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)} + \left(1 + c\right)} \]
   associate-*r/ [=>] 2.07	\[ \frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}} + \left(1 + c\right)} \]
   associate-*l/ [<=] 2.04	\[ \frac{1}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt{\pi}}}{cosTheta} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}\right)} + \left(1 + c\right)} \]
   associate-*r* [=>] 2.04	\[ \frac{1}{\color{blue}{\left(e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{cosTheta}\right) \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}} + \left(1 + c\right)} \]
   *-commutative [=>] 2.04	\[ \frac{1}{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \left(e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{cosTheta}\right)} + \left(1 + c\right)} \]
   fma-def [=>] 2.04	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{\left(1 - cosTheta\right) - cosTheta}, e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{\frac{1}{\sqrt{\pi}}}{cosTheta}, 1 + c\right)}} \]

3. Final simplification 1.47

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

Alternatives

Alternative 1
Error	1.48%
Cost	13376

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

Alternative 2
Error	1.95%
Cost	10208

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

Alternative 3
Error	3.48%
Cost	6976

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

Alternative 4
Error	4.7%
Cost	6848

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

Alternative 5
Error	4.8%
Cost	6784

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

Alternative 6
Error	6.56%
Cost	6464

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

Alternative 7
Error	89.35%
Cost	96

\[1 - c \]

Alternative 8
Error	89.35%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023102 
(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))))))