Beckmann Sample, normalization factor

Average Error: 0.7 → 0.5

Time: 19.5s

Precision: binary32

Cost: 13408

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

Math

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

↓

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

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 (- (- 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(cosTheta * Float32(sqrt(Float32(pi)) / exp(Float32(cosTheta * Float32(-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 0.7

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

4. Simplified 0.5

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

   [Start] 0.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{1}{\sqrt{\pi}}}{\frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{e^{cosTheta \cdot \left(-cosTheta\right)}}}} \]
   rational.json-simplify-61 [<=] 0.7	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{\frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\frac{1}{\sqrt{\pi}}}}}} \]
   rational.json-simplify-61 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{\color{blue}{\frac{\sqrt{\pi}}{\frac{1}{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}}}}} \]
   rational.json-simplify-61 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{1}{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}}{\frac{\sqrt{\pi}}{e^{cosTheta \cdot \left(-cosTheta\right)}}}}} \]
   rational.json-simplify-61 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\frac{cosTheta}{1}}}}{\frac{\sqrt{\pi}}{e^{cosTheta \cdot \left(-cosTheta\right)}}}} \]
   rational.json-simplify-7 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\color{blue}{cosTheta}}}{\frac{\sqrt{\pi}}{e^{cosTheta \cdot \left(-cosTheta\right)}}}} \]
   rational.json-simplify-47 [=>] 0.5	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \frac{\sqrt{\pi}}{e^{cosTheta \cdot \left(-cosTheta\right)}}}}} \]

5. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	13376

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

Alternative 2
Error	0.5
Cost	13376

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

Alternative 3
Error	0.8
Cost	13344

\[\frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\pi} \cdot \left({cosTheta}^{3} + cosTheta\right)}} \]

Alternative 4
Error	0.8
Cost	13312

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

Alternative 5
Error	1.2
Cost	10304

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

Alternative 6
Error	1.2
Cost	6976

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

Alternative 7
Error	1.2
Cost	6912

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

Alternative 8
Error	1.2
Cost	6912

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

Alternative 9
Error	1.6
Cost	6848

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

Alternative 10
Error	1.6
Cost	6784

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

Alternative 11
Error	2.3
Cost	6464

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

Alternative 12
Error	28.6
Cost	96

\[1 - c \]

Alternative 13
Error	28.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))))))