Average Error: 0.1 → 0.1
Time: 10.1s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land -1.5707964 \leq v \land v \leq 0.1\]
\[e^{\left(\left(\left(\frac{cosTheta_i \cdot cosTheta_O}{v} - \frac{sinTheta_i \cdot sinTheta_O}{v}\right) - \frac{1}{v}\right) + 0.6931\right) + \log \left(\frac{1}{2 \cdot v}\right)} \]
↓
\[\frac{{e}^{\left(\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v} + \left(0.6931 - \log v\right)\right)}}{{e}^{\log 2}} \]
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(\frac{\left(cosTheta_i \cdot cosTheta_O - sinTheta_i \cdot sinTheta_O\right) + -1}{v} + \left(0.6931 - \log v\right)\right)}}{{e}^{\log 2}}

(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
   (+
    (/ (+ (- (* cosTheta_i cosTheta_O) (* sinTheta_i sinTheta_O)) -1.0) v)
    (- 0.6931 (log v))))
  (pow E (log 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), (((((cosTheta_i * cosTheta_O) - (sinTheta_i * sinTheta_O)) + -1.0f) / v) + (0.6931f - logf(v)))) / powf(((float) M_E), logf(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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied log-rec_binary320.1

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

  4. Simplified0.1

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

  6. Applied log-prod_binary320.1

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

  7. Applied distribute-neg-in_binary320.1

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

  8. Applied associate-+r+_binary320.1

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

  9. Simplified0.1

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

  11. Applied *-un-lft-identity_binary320.1

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

  12. Applied exp-prod_binary320.1

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

  13. Simplified0.1

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

  15. Applied unsub-neg_binary320.1

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

  16. Applied pow-sub_binary320.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2021204 
(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))))))