Average Error: 0.5 → 0.4
Time: 14.0s
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]) \\]
\[\frac{e^{-\frac{sinTheta_i \cdot sinTheta_O}{v}} \cdot \frac{cosTheta_i \cdot cosTheta_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{1}{v}}\\ cosTheta_O \cdot \left(\frac{cosTheta_i}{v \cdot t_0 - \frac{v}{t_0}} \cdot \frac{\frac{1}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right) \end{array} \]
\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 \left(\frac{cosTheta_i}{v \cdot t_0 - \frac{v}{t_0}} \cdot \frac{\frac{1}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right)
\end{array}

(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) (/ v t_0)))
     (/ (/ 1.0 v) (exp (/ (* sinTheta_i sinTheta_O) 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) {
	float t_0 = expf(1.0f / v);
	return cosTheta_O * ((cosTheta_i / ((v * t_0) - (v / t_0))) * ((1.0f / v) / expf((sinTheta_i * sinTheta_O) / 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 add-exp-log_binary320.4

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

  4. Applied add-exp-log_binary320.5

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

  5. Applied prod-exp_binary320.5

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

  6. Applied prod-exp_binary320.5

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

  7. Applied add-exp-log_binary320.5

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

  8. Applied prod-exp_binary320.6

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

  9. Simplified0.5

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

  10. Applied exp-sum_binary320.5

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

  11. Applied div-inv_binary320.4

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

  12. Applied times-frac_binary320.4

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

  13. Simplified0.4

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

  14. Taylor expanded in cosTheta_i around inf 0.4

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

  15. Simplified0.4

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

  16. Applied *-un-lft-identity_binary320.4

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

  17. Applied *-un-lft-identity_binary320.4

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

  18. Applied add-cube-cbrt_binary320.4

    \[\leadsto cosTheta_O \cdot \left(\left(1 \cdot \frac{cosTheta_i}{v \cdot e^{\frac{1}{v}} - \frac{v}{e^{\frac{1}{v}}}}\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot v}}{1 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right) \]

  19. Applied times-frac_binary320.4

    \[\leadsto cosTheta_O \cdot \left(\left(1 \cdot \frac{cosTheta_i}{v \cdot e^{\frac{1}{v}} - \frac{v}{e^{\frac{1}{v}}}}\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{v}}}{1 \cdot e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right) \]

  20. Applied times-frac_binary320.4

    \[\leadsto cosTheta_O \cdot \left(\left(1 \cdot \frac{cosTheta_i}{v \cdot e^{\frac{1}{v}} - \frac{v}{e^{\frac{1}{v}}}}\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{1} \cdot \frac{\frac{\sqrt[3]{1}}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right)}\right) \]

  21. Applied associate-*r*_binary320.4

    \[\leadsto cosTheta_O \cdot \color{blue}{\left(\left(\left(1 \cdot \frac{cosTheta_i}{v \cdot e^{\frac{1}{v}} - \frac{v}{e^{\frac{1}{v}}}}\right) \cdot \frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right)} \]

  22. Simplified0.4

    \[\leadsto cosTheta_O \cdot \left(\color{blue}{\frac{cosTheta_i}{v \cdot e^{\frac{1}{v}} - \frac{v}{e^{\frac{1}{v}}}}} \cdot \frac{\frac{\sqrt[3]{1}}{v}}{e^{\frac{sinTheta_i \cdot sinTheta_O}{v}}}\right) \]

  23. Final simplification0.4

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

Reproduce

herbie shell --seed 2022094 
(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)))