Beckmann Sample, normalization factor

Average Accuracy: 97.8% → 98.5%

Time: 18.6s

Precision: binary32

Cost: 19680

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

Math

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

↓

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

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
  (+
   1.0
   (+
    c
    (/
     (sqrt (fma cosTheta -2.0 1.0))
     (* (pow (exp cosTheta) cosTheta) (* cosTheta (sqrt PI))))))))

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 / (1.0f + (c + (sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / (powf(expf(cosTheta), cosTheta) * (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) / 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) / Float32(Float32(1.0) + Float32(c + Float32(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / Float32((exp(cosTheta) ^ cosTheta) * Float32(cosTheta * sqrt(Float32(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. Simplified 98.5%

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

   [Start] 97.8	\[ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   associate-+l+ [=>] 97.8	\[ \frac{1}{\color{blue}{1 + \left(c + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]
   *-commutative [=>] 97.8	\[ \frac{1}{1 + \left(c + \color{blue}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}\right)} \]
   associate-*r* [=>] 97.8	\[ \frac{1}{1 + \left(c + \color{blue}{\left(e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}\right)} \]
   *-commutative [<=] 97.8	\[ \frac{1}{1 + \left(c + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot \left(e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{1}{\sqrt{\pi}}\right)}\right)} \]
   associate-/r/ [<=] 97.8	\[ \frac{1}{1 + \left(c + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\frac{cosTheta}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \frac{1}{\sqrt{\pi}}}}}\right)} \]

3. Final simplification 98.5%

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

Alternatives

Alternative 1
Accuracy	98.5%
Cost	16512

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

Alternative 2
Accuracy	98.4%
Cost	13344

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

Alternative 3
Accuracy	98.0%
Cost	13312

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

Alternative 4
Accuracy	97.5%
Cost	10432

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

Alternative 5
Accuracy	97.1%
Cost	10304

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

Alternative 6
Accuracy	97.6%
Cost	10176

\[\frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{1}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}} \]

Alternative 7
Accuracy	96.4%
Cost	6976

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

Alternative 8
Accuracy	95.8%
Cost	6784

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

Alternative 9
Accuracy	93.0%
Cost	6464

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

Alternative 10
Accuracy	10.7%
Cost	96

\[1 - c \]

Alternative 11
Accuracy	10.7%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023147 
(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))))))