Beckmann Sample, normalization factor

Average Error: 0.7 → 0.5

Time: 8.7s

Precision: binary32

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

Math

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

↓

\[\frac{1}{\left(1 + c\right) - \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\left(-\sqrt[3]{\pi}\right) \cdot \left({\pi}^{0.16666666666666666} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)}} \]

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 (fma cosTheta -2.0 1.0))
    (*
     (- (cbrt PI))
     (*
      (pow PI 0.16666666666666666)
      (* cosTheta (pow (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(fmaf(cosTheta, -2.0f, 1.0f)) / (-cbrtf(((float) M_PI)) * (powf(((float) M_PI), 0.16666666666666666f) * (cosTheta * powf(expf(cosTheta), cosTheta))))));
}

Error

Bits error versus cosTheta

Bits error versus c

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 0.5

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

3. Applied egg-rr 0.7

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

4. Applied egg-rr 0.5

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{{\left(\mathsf{fma}\left(cosTheta, -2, 1\right)\right)}^{0.16666666666666666}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}}{\sqrt{\pi}}}} \]

5. Applied egg-rr 0.5

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

6. Applied egg-rr 0.5

   \[\leadsto \frac{1}{\left(1 + c\right) + \color{blue}{\frac{-1 \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\left(-\sqrt[3]{\pi}\right) \cdot \left({\pi}^{0.16666666666666666} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)}}} \]

7. Final simplification 0.5

   \[\leadsto \frac{1}{\left(1 + c\right) - \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\left(-\sqrt[3]{\pi}\right) \cdot \left({\pi}^{0.16666666666666666} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)}} \]

Reproduce

herbie shell --seed 2022130 
(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))))))