Beckmann Sample, normalization factor

Average Error: 0.7 → 1.0

Time: 21.3s

Precision: binary32

Cost: 52384

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := 1 + \left(\left(-t_0\right) + c\right)\\ cosTheta \cdot \sqrt{\pi} + -1 \cdot \left({cosTheta}^{3} \cdot \left(\left(-{t_1}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \pi \cdot \left(t_0 \cdot -1.5\right)\right) + t_1 \cdot \left(\pi \cdot {cosTheta}^{2}\right)\right) \end{array} \]

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 (sqrt (/ 1.0 PI))) (t_1 (+ 1.0 (+ (- t_0) c))))
   (+
    (* cosTheta (sqrt PI))
    (*
     -1.0
     (+
      (*
       (pow cosTheta 3.0)
       (+ (- (* (pow t_1 2.0) (sqrt (pow PI 3.0)))) (* PI (* t_0 -1.5))))
      (* t_1 (* PI (pow cosTheta 2.0))))))))

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 = sqrtf((1.0f / ((float) M_PI)));
	float t_1 = 1.0f + (-t_0 + c);
	return (cosTheta * sqrtf(((float) M_PI))) + (-1.0f * ((powf(cosTheta, 3.0f) * (-(powf(t_1, 2.0f) * sqrtf(powf(((float) M_PI), 3.0f))) + (((float) M_PI) * (t_0 * -1.5f)))) + (t_1 * (((float) M_PI) * powf(cosTheta, 2.0f)))));
}

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 = sqrt(Float32(Float32(1.0) / Float32(pi)))
	t_1 = Float32(Float32(1.0) + Float32(Float32(-t_0) + c))
	return Float32(Float32(cosTheta * sqrt(Float32(pi))) + Float32(Float32(-1.0) * Float32(Float32((cosTheta ^ Float32(3.0)) * Float32(Float32(-Float32((t_1 ^ Float32(2.0)) * sqrt((Float32(pi) ^ Float32(3.0))))) + Float32(Float32(pi) * Float32(t_0 * Float32(-1.5))))) + Float32(t_1 * Float32(Float32(pi) * (cosTheta ^ Float32(2.0)))))))
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 = sqrt((single(1.0) / single(pi)));
	t_1 = single(1.0) + (-t_0 + c);
	tmp = (cosTheta * sqrt(single(pi))) + (single(-1.0) * (((cosTheta ^ single(3.0)) * (-((t_1 ^ single(2.0)) * sqrt((single(pi) ^ single(3.0)))) + (single(pi) * (t_0 * single(-1.5))))) + (t_1 * (single(pi) * (cosTheta ^ single(2.0))))));
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. Taylor expanded in cosTheta around 0 1.0

   \[\leadsto \color{blue}{-1 \cdot \left(\left(-1 \cdot \left({\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \pi\right) \cdot {cosTheta}^{3}\right) + \left(cosTheta \cdot \sqrt{\pi} + -1 \cdot \left({cosTheta}^{2} \cdot \left(\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \pi\right)\right)\right)} \]

3. Simplified 1.0

   \[\leadsto \color{blue}{cosTheta \cdot \sqrt{\pi} + -1 \cdot \left({cosTheta}^{3} \cdot \left(\left(-{\left(1 + \left(\left(-\sqrt{\frac{1}{\pi}}\right) + c\right)\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \pi \cdot \left(\sqrt{\frac{1}{\pi}} \cdot -1.5\right)\right) + \left(1 + \left(\left(-\sqrt{\frac{1}{\pi}}\right) + c\right)\right) \cdot \left(\pi \cdot {cosTheta}^{2}\right)\right)} \]
   Proof

   [Start] 1.0	\[ -1 \cdot \left(\left(-1 \cdot \left({\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \pi\right) \cdot {cosTheta}^{3}\right) + \left(cosTheta \cdot \sqrt{\pi} + -1 \cdot \left({cosTheta}^{2} \cdot \left(\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \pi\right)\right)\right) \]
   rational_best_45_simplify-80 [=>] 1.0	\[ \color{blue}{cosTheta \cdot \sqrt{\pi} + \left(-1 \cdot \left(\left(-1 \cdot \left({\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \pi\right) \cdot {cosTheta}^{3}\right) + -1 \cdot \left({cosTheta}^{2} \cdot \left(\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \pi\right)\right)\right)} \]
   rational_best_45_simplify-91 [=>] 1.0	\[ cosTheta \cdot \sqrt{\pi} + \left(\color{blue}{\left(\left(-1 \cdot \left({\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \left(-1 \cdot \sqrt{\frac{1}{\pi}} + -0.5 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \pi\right) \cdot {cosTheta}^{3}\right) \cdot -1} + -1 \cdot \left({cosTheta}^{2} \cdot \left(\left(c + \left(1 + -1 \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot \pi\right)\right)\right) \]

4. Final simplification 1.0

   \[\leadsto cosTheta \cdot \sqrt{\pi} + -1 \cdot \left({cosTheta}^{3} \cdot \left(\left(-{\left(1 + \left(\left(-\sqrt{\frac{1}{\pi}}\right) + c\right)\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \pi \cdot \left(\sqrt{\frac{1}{\pi}} \cdot -1.5\right)\right) + \left(1 + \left(\left(-\sqrt{\frac{1}{\pi}}\right) + c\right)\right) \cdot \left(\pi \cdot {cosTheta}^{2}\right)\right) \]

Alternatives

Alternative 1
Error	1.0
Cost	52256

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := 1 + \left(-t_0\right)\\ cosTheta \cdot \sqrt{\pi} + -1 \cdot \left({cosTheta}^{2} \cdot \left(\pi \cdot t_1\right) + {cosTheta}^{3} \cdot \left(\left(-{t_1}^{2} \cdot \sqrt{{\pi}^{3}}\right) + \pi \cdot \left(t_0 \cdot -1.5\right)\right)\right) \end{array} \]

Alternative 2
Error	1.0
Cost	48864

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \left(cosTheta \cdot \sqrt{\pi} + {cosTheta}^{2} \cdot \left(\pi \cdot \left(-1 - \left(-t_0\right)\right)\right)\right) + \left(\sqrt{\pi} \cdot -1.5 - {\left(1 - t_0\right)}^{2} \cdot \sqrt{{\pi}^{3}}\right) \cdot \left(-{cosTheta}^{3}\right) \end{array} \]

Alternative 3
Error	0.8
Cost	13312

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

Alternative 4
Error	1.2
Cost	10400

\[\frac{1}{1 + \left(c + \sqrt{\frac{1}{\pi}} \cdot \left(e^{-cosTheta \cdot cosTheta} \cdot \left(\frac{1}{cosTheta} + \left(-1 + cosTheta \cdot -0.5\right)\right)\right)\right)} - 0 \]

Alternative 5
Error	1.2
Cost	10336

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

Alternative 6
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 7
Error	1.6
Cost	6848

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

Alternative 8
Error	1.6
Cost	6784

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

Alternative 9
Error	2.2
Cost	6464

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

Alternative 10
Error	28.6
Cost	128

\[1 + \left(-c\right) \]

Alternative 11
Error	28.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023098 
(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))))))