Average Error: 23.7 → 11.1
Time: 22.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.244272110236542990854135182922651770205 \cdot 10^{212}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right) + \log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\frac{4}{\alpha}}{\alpha}\right)}{2}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.244272110236542990854135182922651770205 \cdot 10^{212}:\\
\;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right) + \log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\frac{4}{\alpha}}{\alpha}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r92957 = alpha;
        double r92958 = beta;
        double r92959 = r92957 + r92958;
        double r92960 = r92958 - r92957;
        double r92961 = r92959 * r92960;
        double r92962 = 2.0;
        double r92963 = i;
        double r92964 = r92962 * r92963;
        double r92965 = r92959 + r92964;
        double r92966 = r92961 / r92965;
        double r92967 = r92965 + r92962;
        double r92968 = r92966 / r92967;
        double r92969 = 1.0;
        double r92970 = r92968 + r92969;
        double r92971 = r92970 / r92962;
        return r92971;
}
↓
double f(double alpha, double beta, double i) {
        double r92972 = alpha;
        double r92973 = 1.244272110236543e+212;
        bool r92974 = r92972 <= r92973;
        double r92975 = beta;
        double r92976 = r92975 + r92972;
        double r92977 = r92975 - r92972;
        double r92978 = 2.0;
        double r92979 = i;
        double r92980 = r92978 * r92979;
        double r92981 = r92972 + r92980;
        double r92982 = r92975 + r92981;
        double r92983 = r92978 + r92982;
        double r92984 = r92977 / r92983;
        double r92985 = r92976 * r92984;
        double r92986 = r92980 + r92976;
        double r92987 = r92985 / r92986;
        double r92988 = 1.0;
        double r92989 = r92987 + r92988;
        double r92990 = exp(r92989);
        double r92991 = sqrt(r92990);
        double r92992 = log(r92991);
        double r92993 = r92992 + r92992;
        double r92994 = r92993 / r92978;
        double r92995 = r92978 / r92972;
        double r92996 = 8.0;
        double r92997 = 3.0;
        double r92998 = pow(r92972, r92997);
        double r92999 = r92996 / r92998;
        double r93000 = 4.0;
        double r93001 = r93000 / r92972;
        double r93002 = r93001 / r92972;
        double r93003 = r92999 - r93002;
        double r93004 = r92995 + r93003;
        double r93005 = r93004 / r92978;
        double r93006 = r92974 ? r92994 : r93005;
        return r93006;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.244272110236543e+212

    1. Initial program 18.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

    2. Simplified18.9

      \[\leadsto \color{blue}{\frac{1 + \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}{2}}\]
    3. Using strategy rm

    4. Applied add-log-exp18.9

      \[\leadsto \frac{1 + \color{blue}{\log \left(e^{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}\right)}}{2}\]

    5. Applied add-log-exp18.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}\right)}{2}\]

    6. Applied sum-log18.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{1} \cdot e^{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}\right)}}{2}\]

    7. Simplified7.3

      \[\leadsto \frac{\log \color{blue}{\left(e^{1 + \frac{\alpha + \beta}{2 \cdot i + \left(\alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \left(\beta + 2\right)\right) + 2 \cdot i}}\right)}}{2}\]
    8. Using strategy rm

    9. Applied div-inv7.3

      \[\leadsto \frac{\log \left(e^{1 + \color{blue}{\left(\left(\alpha + \beta\right) \cdot \frac{1}{2 \cdot i + \left(\alpha + \beta\right)}\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \left(\beta + 2\right)\right) + 2 \cdot i}}\right)}{2}\]

    10. Applied associate-*l*7.3

      \[\leadsto \frac{\log \left(e^{1 + \color{blue}{\left(\alpha + \beta\right) \cdot \left(\frac{1}{2 \cdot i + \left(\alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \left(\beta + 2\right)\right) + 2 \cdot i}\right)}}\right)}{2}\]

    11. Simplified7.3

      \[\leadsto \frac{\log \left(e^{1 + \left(\alpha + \beta\right) \cdot \color{blue}{\frac{\frac{\beta - \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}{2}\]
    12. Using strategy rm

    13. Applied add-sqr-sqrt7.3

      \[\leadsto \frac{\log \color{blue}{\left(\sqrt{e^{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}} \cdot \sqrt{e^{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}}{2}\]

    14. Applied log-prod7.3

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{e^{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}}\right) + \log \left(\sqrt{e^{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}}{2}\]

    15. Simplified7.3

      \[\leadsto \frac{\color{blue}{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\left(\alpha + i \cdot 2\right) + \beta\right) + 2}}{\left(\beta + \alpha\right) + i \cdot 2} + 1}}\right)} + \log \left(\sqrt{e^{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(i \cdot 2 + \alpha\right) + \left(\beta + 2\right)}}{\left(\alpha + \beta\right) + i \cdot 2}}}\right)}{2}\]

    16. Simplified7.3

      \[\leadsto \frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\left(\alpha + i \cdot 2\right) + \beta\right) + 2}}{\left(\beta + \alpha\right) + i \cdot 2} + 1}}\right) + \color{blue}{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\left(\alpha + i \cdot 2\right) + \beta\right) + 2}}{\left(\beta + \alpha\right) + i \cdot 2} + 1}}\right)}}{2}\]

    if 1.244272110236543e+212 < alpha

    1. Initial program 64.0

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

    2. Simplified64.0

      \[\leadsto \color{blue}{\frac{1 + \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}{2}}\]
    3. Using strategy rm

    4. Applied add-log-exp64.0

      \[\leadsto \frac{1 + \color{blue}{\log \left(e^{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}\right)}}{2}\]

    5. Applied add-log-exp64.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}\right)}{2}\]

    6. Applied sum-log64.0

      \[\leadsto \frac{\color{blue}{\log \left(e^{1} \cdot e^{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + 2 \cdot i\right) + \beta\right) \cdot \left(\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i\right)}}\right)}}{2}\]

    7. Simplified50.1

      \[\leadsto \frac{\log \color{blue}{\left(e^{1 + \frac{\alpha + \beta}{2 \cdot i + \left(\alpha + \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \left(\beta + 2\right)\right) + 2 \cdot i}}\right)}}{2}\]

    8. Taylor expanded around inf 42.9

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

    9. Simplified42.9

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\frac{4}{\alpha}}{\alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.244272110236542990854135182922651770205 \cdot 10^{212}:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right) + \log \left(\sqrt{e^{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{2 + \left(\beta + \left(\alpha + 2 \cdot i\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)} + 1}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{\frac{4}{\alpha}}{\alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))