HairBSDF, Mp, upper

Average Error: 0.5 → 0.4

Time: 15.2s

Precision: binary32

\[\left(\left(\left(\left(\left(-1 \leq cosTheta_i \land cosTheta_i \leq 1\right) \land \left(-1 \leq cosTheta_O \land cosTheta_O \leq 1\right)\right) \land \left(-1 \leq sinTheta_i \land sinTheta_i \leq 1\right)\right) \land \left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

\[[cosTheta_i, cosTheta_O] = \mathsf{sort}([cosTheta_i, cosTheta_O]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{1}{v}}\\ cosTheta_O \cdot \frac{cosTheta_i}{\frac{v \cdot t_0 + \left(sinTheta_i \cdot \left(t_0 \cdot sinTheta_O\right) - e^{\frac{-1}{v}} \cdot \mathsf{fma}\left(sinTheta_i, sinTheta_O, v\right)\right)}{\frac{1}{v}}} \end{array} \]

FPCore

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

↓

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (let* ((t_0 (exp (/ 1.0 v))))
   (*
    cosTheta_O
    (/
     cosTheta_i
     (/
      (+
       (* v t_0)
       (-
        (* sinTheta_i (* t_0 sinTheta_O))
        (* (exp (/ -1.0 v)) (fma sinTheta_i sinTheta_O v))))
      (/ 1.0 v))))))

C

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

↓

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	float t_0 = expf(1.0f / v);
	return cosTheta_O * (cosTheta_i / (((v * t_0) + ((sinTheta_i * (t_0 * sinTheta_O)) - (expf(-1.0f / v) * fmaf(sinTheta_i, sinTheta_O, v)))) / (1.0f / 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 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \left(\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)}} \]

3. Applied add-exp-log_binary32 0.4

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \left(\left(\sinh \left(\frac{1}{v}\right) \cdot \color{blue}{e^{\log 2}}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \]

4. Applied add-exp-log_binary32 0.5

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \left(\left(\color{blue}{e^{\log \sinh \left(\frac{1}{v}\right)}} \cdot e^{\log 2}\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \]

5. Applied prod-exp_binary32 0.5

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \left(\color{blue}{e^{\log \sinh \left(\frac{1}{v}\right) + \log 2}} \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)} \]

6. Applied prod-exp_binary32 0.5

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{v \cdot \color{blue}{e^{\left(\log \sinh \left(\frac{1}{v}\right) + \log 2\right) + \frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

7. Applied add-exp-log_binary32 0.5

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{\color{blue}{e^{\log v}} \cdot e^{\left(\log \sinh \left(\frac{1}{v}\right) + \log 2\right) + \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

8. Applied prod-exp_binary32 0.6

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{\color{blue}{e^{\log v + \left(\left(\log \sinh \left(\frac{1}{v}\right) + \log 2\right) + \frac{sinTheta_i \cdot sinTheta_O}{v}\right)}}} \]

9. Simplified 0.5

   \[\leadsto cosTheta_O \cdot \frac{\frac{cosTheta_i}{v}}{e^{\color{blue}{\log \left(v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)\right) + \frac{sinTheta_i \cdot sinTheta_O}{v}}}} \]

10. Applied div-inv_binary32 0.4

    \[\leadsto cosTheta_O \cdot \frac{\color{blue}{cosTheta_i \cdot \frac{1}{v}}}{e^{\log \left(v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)\right) + \frac{sinTheta_i \cdot sinTheta_O}{v}}} \]

11. Applied associate-/l*_binary32 0.4

    \[\leadsto cosTheta_O \cdot \color{blue}{\frac{cosTheta_i}{\frac{e^{\log \left(v \cdot \left(2 \cdot \sinh \left(\frac{1}{v}\right)\right)\right) + \frac{sinTheta_i \cdot sinTheta_O}{v}}}{\frac{1}{v}}}} \]

12. Taylor expanded in sinTheta_i around 0 0.4

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

13. Simplified 0.4

    \[\leadsto cosTheta_O \cdot \frac{cosTheta_i}{\frac{\color{blue}{\mathsf{fma}\left(v, e^{\frac{1}{v}}, sinTheta_i \cdot \left(sinTheta_O \cdot e^{\frac{1}{v}}\right)\right) - \left(\frac{v}{e^{\frac{1}{v}}} + \frac{sinTheta_i \cdot sinTheta_O}{e^{\frac{1}{v}}}\right)}}{\frac{1}{v}}} \]

14. Applied fma-udef_binary32 0.4

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

15. Applied associate--l+_binary32 0.4

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

16. Simplified 0.4

    \[\leadsto cosTheta_O \cdot \frac{cosTheta_i}{\frac{v \cdot e^{\frac{1}{v}} + \color{blue}{\left(sinTheta_i \cdot \left(e^{\frac{1}{v}} \cdot sinTheta_O\right) - e^{\frac{-1}{v}} \cdot \mathsf{fma}\left(sinTheta_i, sinTheta_O, v\right)\right)}}{\frac{1}{v}}} \]

17. Final simplification 0.4

    \[\leadsto cosTheta_O \cdot \frac{cosTheta_i}{\frac{v \cdot e^{\frac{1}{v}} + \left(sinTheta_i \cdot \left(e^{\frac{1}{v}} \cdot sinTheta_O\right) - e^{\frac{-1}{v}} \cdot \mathsf{fma}\left(sinTheta_i, sinTheta_O, v\right)\right)}{\frac{1}{v}}} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))