Average Error: 0.1 → 0.1
Time: 8.6s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land -1.5707964 \leq v \land v \leq 0.1\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\frac{{e}^{\left(0.6931 + \left(\mathsf{fma}\left(\frac{cosTheta_i}{v}, cosTheta_O, \frac{-1}{v}\right) - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right)}}{{e}^{\log \left(v \cdot 2\right)}} \]
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)}
↓
\frac{{e}^{\left(0.6931 + \left(\mathsf{fma}\left(\frac{cosTheta_i}{v}, cosTheta_O, \frac{-1}{v}\right) - \frac{sinTheta_i \cdot sinTheta_O}{v}\right)\right)}}{{e}^{\log \left(v \cdot 2\right)}}

(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
    (-
     (fma (/ cosTheta_i v) cosTheta_O (/ -1.0 v))
     (/ (* sinTheta_i sinTheta_O) v))))
  (pow E (log (* v 2.0)))))
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 + (fmaf((cosTheta_i / v), cosTheta_O, (-1.0f / v)) - ((sinTheta_i * sinTheta_O) / v)))) / powf(((float) M_E), logf(v * 2.0f));
}

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. Applied *-un-lft-identity_binary320.1

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

  3. Applied exp-prod_binary320.1

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

  4. Simplified0.1

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

  5. Applied log-rec_binary320.1

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

  6. Applied unsub-neg_binary320.1

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

  7. Applied pow-sub_binary320.1

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

  8. Simplified0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2021224 
(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))))))