Average Error: 0.5 → 0.4
Time: 17.8s
Precision: binary32
\[-1 \leq cosTheta_i \land cosTheta_i \leq 1 \land -1 \leq cosTheta_O \land cosTheta_O \leq 1 \land -1 \leq sinTheta_i \land sinTheta_i \leq 1 \land -1 \leq sinTheta_O \land sinTheta_O \leq 1 \land 0.1 < v \land v \leq 1.5707964\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[cosTheta_O \cdot \frac{cosTheta_i}{\frac{v \cdot \left(\left(\left(e^{\frac{1}{v}} - \log \left(e^{e^{\frac{-1}{v}}}\right)\right) \cdot 2\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)}{\frac{2}{v}}} \]
\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}
↓
cosTheta_O \cdot \frac{cosTheta_i}{\frac{v \cdot \left(\left(\left(e^{\frac{1}{v}} - \log \left(e^{e^{\frac{-1}{v}}}\right)\right) \cdot 2\right) \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}\right)}{\frac{2}{v}}}

(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
 (*
  cosTheta_O
  (/
   cosTheta_i
   (/
    (*
     v
     (*
      (* (- (exp (/ 1.0 v)) (log (exp (exp (/ -1.0 v))))) 2.0)
      (exp (/ (* sinTheta_i sinTheta_O) v))))
    (/ 2.0 v)))))
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) {
	return cosTheta_O * (cosTheta_i / ((v * (((expf(1.0f / v) - logf(expf(expf(-1.0f / v)))) * 2.0f) * expf((sinTheta_i * sinTheta_O) / v))) / (2.0f / 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.4

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

  3. Applied associate-/l/_binary320.4

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

  4. Applied sinh-def_binary320.4

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

  5. Applied associate-*l/_binary320.4

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

  6. Applied associate-*l/_binary320.4

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

  7. Applied associate-*r/_binary320.4

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

  8. Applied associate-*l/_binary320.4

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

  9. Applied associate-/l*_binary320.4

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

  10. Applied add-log-exp_binary320.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2021215 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :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) (< 0.1 v) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))