Beckmann Sample, normalization factor

Average Error: 0.7 → 0.5

Time: 18.1s

Precision: binary32

Cost: 13376

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

Math

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

↓

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

FPCore

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

↓

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

↓

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

↓

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

↓

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

Derivation

1. Initial program 0.7

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

2. Simplified 0.7

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

   [Start] 0.7	\[ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-2 [=>] 0.7	\[ \frac{1}{\left(1 + c\right) + \color{blue}{e^{\left(-cosTheta\right) \cdot cosTheta} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right)}} \]
   rational.json-simplify-43 [=>] 0.7	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]
   rational.json-simplify-2 [=>] 0.7	\[ \frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\pi}} \cdot \left(\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta} \cdot e^{\color{blue}{cosTheta \cdot \left(-cosTheta\right)}}\right)} \]

3. Applied egg-rr 0.5

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{e^{cosTheta \cdot cosTheta}}}{\sqrt{\pi}} + 0\right)}} \]

4. Simplified 0.5

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

   [Start] 0.5	\[ \frac{1}{\left(1 + c\right) + \left(\frac{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{e^{cosTheta \cdot cosTheta}}}{\sqrt{\pi}} + 0\right)} \]
   rational.json-simplify-4 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{e^{cosTheta \cdot cosTheta}}}{\sqrt{\pi}}}} \]
   rational.json-simplify-44 [<=] 0.5	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}}} \]
   rational.json-simplify-47 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\pi}}}}{e^{cosTheta \cdot cosTheta}}} \]
   rational.json-simplify-47 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\left(cosTheta \cdot \sqrt{\pi}\right) \cdot e^{cosTheta \cdot cosTheta}}}} \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.8
Cost	13312

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

Alternative 2
Error	1.0
Cost	10272

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

Alternative 3
Error	1.1
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 4
Error	1.2
Cost	6912

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

Alternative 5
Error	1.5
Cost	6848

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

Alternative 6
Error	1.5
Cost	6848

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

Alternative 7
Error	1.6
Cost	6784

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

Alternative 8
Error	1.6
Cost	6784

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

Alternative 9
Error	2.1
Cost	6464

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

Alternative 10
Error	28.6
Cost	96

\[1 - c \]

Alternative 11
Error	28.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023069 
(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))))))