HairBSDF, Mp, lower

Average Error: 0.1 → 0.1

Time: 14.4s

Precision: binary32

\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land -1.5707964 \leq v \land v \leq 0.1\]

Math

\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{v}}\\ t_1 := \sqrt[3]{t_0}\\ t_2 := \sqrt[3]{{t_1}^{5}}\\ \frac{1}{{\left(t_1 \cdot t_1\right)}^{5} \cdot \left(t_0 \cdot \left(t_2 \cdot \left(t_2 \cdot t_2\right)\right)\right)} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (exp
  (+
   (+
    (-
     (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v))
     (/ 1.0 v))
    0.6931)
   (log (/ 1.0 (* 2.0 v))))))

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (cbrt (cbrt v))) (t_1 (cbrt t_0)) (t_2 (cbrt (pow t_1 5.0))))
   (*
    (/ 1.0 (* (pow (* t_1 t_1) 5.0) (* t_0 (* t_2 (* t_2 t_2)))))
    (*
     (exp
      (+
       0.6931
       (-
        (/ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) v)
        (/ 1.0 v))))
     (/ 0.5 (cbrt v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return expf((((((cosTheta_i * cosTheta_O) / v) - ((sinTheta_i * sinTheta_O) / v)) - (1.0f / v)) + 0.6931f) + logf(1.0f / (2.0f * v)));
}

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = cbrtf(cbrtf(v));
	float t_1 = cbrtf(t_0);
	float t_2 = cbrtf(powf(t_1, 5.0f));
	return (1.0f / (powf((t_1 * t_1), 5.0f) * (t_0 * (t_2 * (t_2 * t_2))))) * (expf(0.6931f + ((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) / v) - (1.0f / v))) * (0.5f / cbrtf(v)));
}

Error

Bits error versus cosTheta_i

Bits error versus cosTheta_O

Bits error versus sinTheta_i

Bits error versus sinTheta_O

Bits error versus v

Derivation

1. Initial program 0.1

   \[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{0.5}{v} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}} \]
3. Using strategy rm

4. Applied add-cube-cbrt_binary32 0.1

   \[\leadsto \frac{0.5}{\color{blue}{\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \sqrt[3]{v}}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

5. Applied *-un-lft-identity_binary32 0.1

   \[\leadsto \frac{\color{blue}{1 \cdot 0.5}}{\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \sqrt[3]{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

6. Applied times-frac_binary32 0.1

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}} \cdot \frac{0.5}{\sqrt[3]{v}}\right)} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931} \]

7. Applied associate-*l*_binary32 0.1

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}} \cdot \left(\frac{0.5}{\sqrt[3]{v}} \cdot e^{\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931}\right)} \]

8. Simplified 0.1

   \[\leadsto \frac{1}{\sqrt[3]{v} \cdot \sqrt[3]{v}} \cdot \color{blue}{\left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right)} \]
9. Using strategy rm

10. Applied add-cube-cbrt_binary32 0.1

    \[\leadsto \frac{1}{\sqrt[3]{v} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{v}} \cdot \sqrt[3]{\sqrt[3]{v}}\right) \cdot \sqrt[3]{\sqrt[3]{v}}\right)}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

11. Applied associate-*r*_binary32 0.1

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{v} \cdot \left(\sqrt[3]{\sqrt[3]{v}} \cdot \sqrt[3]{\sqrt[3]{v}}\right)\right) \cdot \sqrt[3]{\sqrt[3]{v}}}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

12. Simplified 0.1

    \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{v}}\right)}^{5}} \cdot \sqrt[3]{\sqrt[3]{v}}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]
13. Using strategy rm

14. Applied add-cube-cbrt_binary32 0.1

    \[\leadsto \frac{1}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}}^{5} \cdot \sqrt[3]{\sqrt[3]{v}}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

15. Applied unpow-prod-down_binary32 0.1

    \[\leadsto \frac{1}{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}\right)} \cdot \sqrt[3]{\sqrt[3]{v}}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

16. Applied associate-*l*_binary32 0.1

    \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{v}}\right)}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

17. Simplified 0.1

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{v}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}\right)}} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]
18. Using strategy rm

19. Applied add-cube-cbrt_binary32 0.1

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5} \cdot \left(\sqrt[3]{\sqrt[3]{v}} \cdot \color{blue}{\left(\left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}}\right)}\right)} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

20. Final simplification 0.1

    \[\leadsto \frac{1}{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5} \cdot \left(\sqrt[3]{\sqrt[3]{v}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{v}}}\right)}^{5}}\right)\right)\right)} \cdot \left(e^{0.6931 + \left(\frac{cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O}{v} - \frac{1}{v}\right)} \cdot \frac{0.5}{\sqrt[3]{v}}\right) \]

Reproduce

herbie shell --seed 2021190 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (and (<= -1.0 cosTheta_i 1.0) (<= -1.0 cosTheta_O 1.0) (<= -1.0 sinTheta_i 1.0) (<= -1.0 sinTheta_O 1.0) (<= -1.5707964 v 0.1))
  (exp (+ (+ (- (- (/ (* cosTheta_i cosTheta_O) v) (/ (* sinTheta_i sinTheta_O) v)) (/ 1.0 v)) 0.6931) (log (/ 1.0 (* 2.0 v))))))