Beckmann Sample, normalization factor

Average Error: 0.7 → 0.5

Time: 21.5s

Precision: binary32

Cost: 14016

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

Math

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

↓

\[\begin{array}{l} t_0 := cosTheta - \left(1 - cosTheta\right)\\ \frac{1}{\left(1 + c\right) + \frac{\frac{\frac{\sqrt{t_0 \cdot t_0}}{cosTheta}}{\sqrt{\pi \cdot \left(1 - \left(cosTheta + cosTheta\right)\right)}}}{\left(-cosTheta\right) \cdot \frac{-1}{cosTheta}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \end{array} \]

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
 (let* ((t_0 (- cosTheta (- 1.0 cosTheta))))
   (/
    1.0
    (+
     (+ 1.0 c)
     (*
      (/
       (/
        (/ (sqrt (* t_0 t_0)) cosTheta)
        (sqrt (* PI (- 1.0 (+ cosTheta cosTheta)))))
       (* (- cosTheta) (/ -1.0 cosTheta)))
      (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) {
	float t_0 = cosTheta - (1.0f - cosTheta);
	return 1.0f / ((1.0f + c) + ((((sqrtf((t_0 * t_0)) / cosTheta) / sqrtf((((float) M_PI) * (1.0f - (cosTheta + cosTheta))))) / (-cosTheta * (-1.0f / cosTheta))) * 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)
	t_0 = Float32(cosTheta - Float32(Float32(1.0) - cosTheta))
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(sqrt(Float32(t_0 * t_0)) / cosTheta) / sqrt(Float32(Float32(pi) * Float32(Float32(1.0) - Float32(cosTheta + cosTheta))))) / Float32(Float32(-cosTheta) * Float32(Float32(-1.0) / cosTheta))) * 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)
	t_0 = cosTheta - (single(1.0) - cosTheta);
	tmp = single(1.0) / ((single(1.0) + c) + ((((sqrt((t_0 * t_0)) / cosTheta) / sqrt((single(pi) * (single(1.0) - (cosTheta + cosTheta))))) / (-cosTheta * (single(-1.0) / cosTheta))) * 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 12.7

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

3. Simplified 12.7

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

   [Start] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\left(\sqrt{\pi} \cdot 2\right) \cdot \left(cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}\right)} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-46 [=>] 12.7	\[ \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\pi} \cdot 2}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-47 [=>] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \left(\sqrt{\pi} \cdot 2\right)}}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   metadata-eval [<=] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \left(\sqrt{\pi} \cdot \color{blue}{\left(1 + 1\right)}\right)}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-51 [<=] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \color{blue}{\left(\sqrt{\pi} \cdot 1 + 1 \cdot \sqrt{\pi}\right)}}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-2 [=>] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \left(\color{blue}{1 \cdot \sqrt{\pi}} + 1 \cdot \sqrt{\pi}\right)}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-6 [=>] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \left(\color{blue}{\sqrt{\pi}} + 1 \cdot \sqrt{\pi}\right)}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-6 [=>] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \left(\sqrt{\pi} + \color{blue}{\sqrt{\pi}}\right)}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-51 [<=] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \sqrt{\pi} + \sqrt{\pi} \cdot \sqrt{\left(1 - cosTheta\right) - cosTheta}}}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-2 [<=] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\frac{cosTheta + cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \sqrt{\pi} + \color{blue}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \sqrt{\pi}}}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   rational.json-simplify-35 [<=] 12.7	\[ \frac{1}{\left(1 + c\right) + \frac{\color{blue}{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta} \cdot \sqrt{\pi}}}}{cosTheta \cdot \frac{\frac{cosTheta}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\sqrt{\left(1 - cosTheta\right) - cosTheta}}} \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

4. Applied egg-rr 14.2

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

5. Applied egg-rr 0.5

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

6. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.6
Cost	13376

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

Alternative 2
Error	0.5
Cost	13376

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

Alternative 3
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 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	6976

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

Alternative 6
Error	1.6
Cost	6848

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

Alternative 7
Error	1.6
Cost	6784

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

Alternative 8
Error	2.2
Cost	6464

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

Alternative 9
Error	28.6
Cost	96

\[1 - c \]

Alternative 10
Error	28.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023068 
(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))))))