HairBSDF, Mp, lower

Average Error: 0.1 → 0.2

Time: 9.3s

Precision: binary32

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

Math

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

↓

\[{e}^{\left(\left(\left(0.6931 + \left(\log 0.5 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}\right) + \log \left(\frac{1}{v}\right)\right)} \]

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
 (pow
  E
  (+
   (- (+ 0.6931 (+ (log 0.5) (/ (* cosTheta_i cosTheta_O) v))) (/ 1.0 v))
   (log (/ 1.0 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) {
	return powf(((float) M_E), (((0.6931f + (logf(0.5f) + ((cosTheta_i * cosTheta_O) / v))) - (1.0f / v)) + logf((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 0.1

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

2. Simplified 0.1

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

3. Applied add-exp-log_binary32 0.1

   \[\leadsto \frac{0.5}{\color{blue}{e^{\log v}}} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)} \]

4. Applied add-exp-log_binary32 0.1

   \[\leadsto \frac{\color{blue}{e^{\log 0.5}}}{e^{\log v}} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)} \]

5. Applied div-exp_binary32 0.1

   \[\leadsto \color{blue}{e^{\log 0.5 - \log v}} \cdot e^{\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)} \]

6. Applied prod-exp_binary32 0.1

   \[\leadsto \color{blue}{e^{\left(\log 0.5 - \log v\right) + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)}} \]

7. Simplified 0.1

   \[\leadsto e^{\color{blue}{\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)}} \]

8. Applied *-un-lft-identity_binary32 0.1

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

9. Applied exp-prod_binary32 0.1

   \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)\right)}} \]

10. Simplified 0.1

    \[\leadsto {\color{blue}{e}}^{\left(\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \log \left(\frac{0.5}{v}\right)\right)} \]

11. Applied div-inv_binary32 0.2

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

12. Applied log-prod_binary32 0.2

    \[\leadsto {e}^{\left(\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \color{blue}{\left(\log 0.5 + \log \left(\frac{1}{v}\right)\right)}\right)} \]

13. Applied associate-+r+_binary32 0.2

    \[\leadsto {e}^{\color{blue}{\left(\left(\left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right) + \log 0.5\right) + \log \left(\frac{1}{v}\right)\right)}} \]

14. Simplified 0.2

    \[\leadsto {e}^{\left(\color{blue}{\left(\log 0.5 + \left(\mathsf{fma}\left(cosTheta_O, \frac{cosTheta_i}{v}, 0.6931\right) - \mathsf{fma}\left(sinTheta_i, \frac{sinTheta_O}{v}, \frac{1}{v}\right)\right)\right)} + \log \left(\frac{1}{v}\right)\right)} \]

15. Taylor expanded in sinTheta_i around 0 0.2

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

16. Final simplification 0.2

    \[\leadsto {e}^{\left(\left(\left(0.6931 + \left(\log 0.5 + \frac{cosTheta_i \cdot cosTheta_O}{v}\right)\right) - \frac{1}{v}\right) + \log \left(\frac{1}{v}\right)\right)} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, lower"
  :precision binary32
  :pre (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))) (and (<= -1.5707964 v) (<= 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))))))