Beckmann Sample, normalization factor

Average Error: 0.7 → 0.5

Time: 18.9s

Precision: binary32

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

Math

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

↓

\[\frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{1}{\sqrt{\sqrt[3]{\pi}} \cdot \mathsf{fma}\left({cosTheta}^{3}, \sqrt[3]{\pi}, \mathsf{fma}\left(0.5, \sqrt[3]{\pi} \cdot {cosTheta}^{5}, \mathsf{fma}\left(cosTheta, \sqrt[3]{\pi}, 0.16666666666666666 \cdot \left(\sqrt[3]{\pi} \cdot {cosTheta}^{7}\right)\right)\right)\right)}} \]

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))
    (/
     1.0
     (*
      (sqrt (cbrt PI))
      (fma
       (pow cosTheta 3.0)
       (cbrt PI)
       (fma
        0.5
        (* (cbrt PI) (pow cosTheta 5.0))
        (fma
         cosTheta
         (cbrt PI)
         (* 0.16666666666666666 (* (cbrt PI) (pow cosTheta 7.0))))))))))))

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) {
	return 1.0f / ((1.0f + c) + (sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) * (1.0f / (sqrtf(cbrtf((float) M_PI)) * fmaf(powf(cosTheta, 3.0f), cbrtf((float) M_PI), fmaf(0.5f, (cbrtf((float) M_PI) * powf(cosTheta, 5.0f)), fmaf(cosTheta, cbrtf((float) M_PI), (0.16666666666666666f * (cbrtf((float) M_PI) * powf(cosTheta, 7.0f))))))))));
}

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 add-cube-cbrt_binary32 0.5

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

4. Applied sqrt-prod_binary32 0.5

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

5. Applied associate-*l*_binary32 0.4

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

6. Applied div-inv_binary32 0.5

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

7. Applied sqrt-prod_binary32 1.1

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

8. Applied associate-*l*_binary32 1.2

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

9. Simplified 0.5

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

10. Taylor expanded in cosTheta around 0 0.5

    \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{1}{\sqrt{\sqrt[3]{\pi}} \cdot \color{blue}{\left({\pi}^{0.3333333333333333} \cdot {cosTheta}^{3} + \left(0.5 \cdot \left({\pi}^{0.3333333333333333} \cdot {cosTheta}^{5}\right) + \left({\pi}^{0.3333333333333333} \cdot cosTheta + 0.16666666666666666 \cdot \left({\pi}^{0.3333333333333333} \cdot {cosTheta}^{7}\right)\right)\right)\right)}}} \]

11. Simplified 0.5

    \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{1}{\sqrt{\sqrt[3]{\pi}} \cdot \color{blue}{\mathsf{fma}\left({cosTheta}^{3}, \sqrt[3]{\pi}, \mathsf{fma}\left(0.5, {cosTheta}^{5} \cdot \sqrt[3]{\pi}, \mathsf{fma}\left(cosTheta, \sqrt[3]{\pi}, 0.16666666666666666 \cdot \left({cosTheta}^{7} \cdot \sqrt[3]{\pi}\right)\right)\right)\right)}}} \]

12. Final simplification 0.5

    \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \frac{1}{\sqrt{\sqrt[3]{\pi}} \cdot \mathsf{fma}\left({cosTheta}^{3}, \sqrt[3]{\pi}, \mathsf{fma}\left(0.5, \sqrt[3]{\pi} \cdot {cosTheta}^{5}, \mathsf{fma}\left(cosTheta, \sqrt[3]{\pi}, 0.16666666666666666 \cdot \left(\sqrt[3]{\pi} \cdot {cosTheta}^{7}\right)\right)\right)\right)}} \]

Reproduce

herbie shell --seed 2022068 
(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))))))