Beckmann Sample, normalization factor

Average Error: 0.7 → 0.4

Time: 1.1min

Precision: binary32

Cost: 23200

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

Math

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

↓

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

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)
   (*
    (/
     (* 2.0 (sqrt (+ 1.0 (* cosTheta -2.0))))
     (* (sqrt (* PI PI)) (* 2.0 (/ 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) + (((2.0f * sqrtf((1.0f + (cosTheta * -2.0f)))) / (sqrtf((((float) M_PI) * ((float) M_PI))) * (2.0f * (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(Float32(Float32(Float32(2.0) * sqrt(Float32(Float32(1.0) + Float32(cosTheta * Float32(-2.0))))) / Float32(sqrt(Float32(Float32(pi) * Float32(pi))) * Float32(Float32(2.0) * Float32(cosTheta / sqrt(Float32(pi)))))) * 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

↓

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(2.0) * sqrt((single(1.0) + (cosTheta * single(-2.0))))) / (sqrt((single(pi) * single(pi))) * (single(2.0) * (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. Applied egg-rr 0.5

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

3. Applied egg-rr 0.5

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

4. Simplified 0.4

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

   [Start] 0.5	\[ \frac{1}{\left(1 + c\right) + \left(\frac{\sqrt{1 + cosTheta \cdot -2} \cdot 2}{\sqrt{\pi \cdot \pi}} \cdot \frac{1}{\frac{cosTheta + cosTheta}{\sqrt{\pi}}}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-55 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\left(1 \cdot \frac{\frac{\sqrt{1 + cosTheta \cdot -2} \cdot 2}{\sqrt{\pi \cdot \pi}}}{\frac{cosTheta + cosTheta}{\sqrt{\pi}}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-1 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\left(\frac{\frac{\sqrt{1 + cosTheta \cdot -2} \cdot 2}{\sqrt{\pi \cdot \pi}}}{\frac{cosTheta + cosTheta}{\sqrt{\pi}}} \cdot 1\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-7 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{\sqrt{1 + cosTheta \cdot -2} \cdot 2}{\sqrt{\pi \cdot \pi}}}{\frac{cosTheta + cosTheta}{\sqrt{\pi}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-53 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt{1 + cosTheta \cdot -2} \cdot 2}{\sqrt{\pi \cdot \pi} \cdot \frac{cosTheta + cosTheta}{\sqrt{\pi}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-1 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{\color{blue}{2 \cdot \sqrt{1 + cosTheta \cdot -2}}}{\sqrt{\pi \cdot \pi} \cdot \frac{cosTheta + cosTheta}{\sqrt{\pi}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-65 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \color{blue}{\left(\frac{cosTheta}{\sqrt{\pi}} + \frac{cosTheta}{\sqrt{\pi}}\right)}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-7 [<=] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \left(\color{blue}{\frac{cosTheta}{\sqrt{\pi}} \cdot 1} + \frac{cosTheta}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-1 [<=] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \left(\color{blue}{1 \cdot \frac{cosTheta}{\sqrt{\pi}}} + \frac{cosTheta}{\sqrt{\pi}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-7 [<=] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \left(1 \cdot \frac{cosTheta}{\sqrt{\pi}} + \color{blue}{\frac{cosTheta}{\sqrt{\pi}} \cdot 1}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-1 [<=] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \left(1 \cdot \frac{cosTheta}{\sqrt{\pi}} + \color{blue}{1 \cdot \frac{cosTheta}{\sqrt{\pi}}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-63 [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \color{blue}{\left(\frac{cosTheta}{\sqrt{\pi}} \cdot \left(1 + 1\right)\right)}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   metadata-eval [=>] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \left(\frac{cosTheta}{\sqrt{\pi}} \cdot \color{blue}{2}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational_best-simplify-1 [<=] 0.4	\[ \frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{cosTheta}{\sqrt{\pi}}\right)}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	16768

\[\frac{1}{\left(1 + c\right) + \frac{2 \cdot \sqrt{1 + cosTheta \cdot -2}}{\frac{cosTheta \cdot \pi}{\sqrt{\pi} \cdot 0.5}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

Alternative 2
Error	0.5
Cost	13408

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

Alternative 3
Error	0.5
Cost	13376

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

Alternative 4
Error	0.8
Cost	13312

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

Alternative 5
Error	1.0
Cost	6976

\[\frac{1}{c + \left(1 + \left(\frac{1}{cosTheta} - cosTheta\right) \cdot \sqrt{\left(0.5 - cosTheta\right) \cdot \frac{2}{\pi}}\right)} \]

Alternative 6
Error	1.1
Cost	6912

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

Alternative 7
Error	1.5
Cost	6880

\[\frac{1}{\left(1 + c\right) + \frac{\sqrt{\frac{2}{\pi} \cdot \left(\left(-cosTheta\right) + 0.5\right)}}{cosTheta}} \]

Alternative 8
Error	1.5
Cost	6848

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

Alternative 9
Error	1.6
Cost	6784

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

Alternative 10
Error	2.3
Cost	6464

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

Alternative 11
Error	28.5
Cost	96

\[1 - c \]

Alternative 12
Error	28.5
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023099 
(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))))))