Beckmann Sample, normalization factor

Average Error: 97.8% → 98.6%

Time: 32.0s

Precision: binary32

Cost: 23360.00

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

Math

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

↓

\[\begin{array}{l} t_0 := cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\\ \frac{1}{\left(1 - c \cdot c\right) \cdot t_0 + \left(1 - c\right) \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}} \cdot \left(t_0 \cdot \left(1 - c\right)\right) \end{array} \]

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 (* cosTheta (pow (exp cosTheta) cosTheta))))
   (*
    (/
     1.0
     (+
      (* (- 1.0 (* c c)) t_0)
      (* (- 1.0 c) (sqrt (/ (fma cosTheta -2.0 1.0) PI)))))
    (* t_0 (- 1.0 c)))))

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 * powf(expf(cosTheta), cosTheta);
	return (1.0f / (((1.0f - (c * c)) * t_0) + ((1.0f - c) * sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI)))))) * (t_0 * (1.0f - c));
}

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 * (exp(cosTheta) ^ cosTheta))
	return Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) - Float32(c * c)) * t_0) + Float32(Float32(Float32(1.0) - c) * sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi)))))) * Float32(t_0 * Float32(Float32(1.0) - c)))
end

Derivation

1. Initial program 97.8

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

2. Simplified 98.5

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

   [Start] 97.8	\[ \frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   +-commutative [=>] 97.8	\[ \frac{1}{\color{blue}{\left(c + 1\right)} + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
   associate-+l+ [=>] 97.8	\[ \frac{1}{\color{blue}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}\right)}} \]
   distribute-lft-neg-out [=>] 97.8	\[ \frac{1}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\color{blue}{-cosTheta \cdot cosTheta}}\right)} \]
   exp-neg [=>] 97.8	\[ \frac{1}{c + \left(1 + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot \color{blue}{\frac{1}{e^{cosTheta \cdot cosTheta}}}\right)} \]
   associate-*r/ [=>] 97.8	\[ \frac{1}{c + \left(1 + \color{blue}{\frac{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot 1}{e^{cosTheta \cdot cosTheta}}}\right)} \]
   associate-/l* [=>] 97.8	\[ \frac{1}{c + \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}}\right)} \]
   associate-*l/ [=>] 98.4	\[ \frac{1}{c + \left(1 + \frac{\color{blue}{\frac{1 \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt{\pi}}}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}\right)} \]
   *-lft-identity [=>] 98.4	\[ \frac{1}{c + \left(1 + \frac{\frac{\color{blue}{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}}{\sqrt{\pi}}}{\frac{e^{cosTheta \cdot cosTheta}}{1}}\right)} \]

3. Applied egg-rr 61.2

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

4. Simplified 61.2

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

   [Start] 61.2	\[ \sqrt{{\left(c + \left(1 + \frac{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right)\right)}^{-2}} \]
   +-commutative [=>] 61.2	\[ \sqrt{{\color{blue}{\left(\left(1 + \frac{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}\right) + c\right)}}^{-2}} \]
   +-commutative [=>] 61.2	\[ \sqrt{{\left(\color{blue}{\left(\frac{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}} + 1\right)} + c\right)}^{-2}} \]
   associate-+l+ [=>] 61.2	\[ \sqrt{{\color{blue}{\left(\frac{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}} + \left(1 + c\right)\right)}}^{-2}} \]
   associate-/r* [<=] 61.2	\[ \sqrt{{\left(\color{blue}{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}} + \left(1 + c\right)\right)}^{-2}} \]

5. Applied egg-rr 98.6

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

6. Final simplification 98.6

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

Alternatives

Alternative 1
Error	98.5%
Cost	22848.00

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

Alternative 2
Error	98.5%
Cost	13376.00

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

Alternative 3
Error	98.1%
Cost	10272.00

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

Alternative 4
Error	98.1%
Cost	10208.00

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

Alternative 5
Error	96.3%
Cost	6976.00

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

Alternative 6
Error	94.9%
Cost	6784.00

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

Alternative 7
Error	95.5%
Cost	6784.00

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

Alternative 8
Error	93.1%
Cost	6464.00

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

Alternative 9
Error	10.7%
Cost	32.00

\[1 \]

Reproduce

herbie shell --seed 2023093 
(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))))))