Average Error: 24.0 → 13.1
Time: 44.8s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.76982660809475993097637758871956020598 \cdot 10^{62}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(i \cdot 2 + \alpha\right) + \beta} \cdot \left(\beta - \alpha\right)\right)}{2 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{2}\\ \mathbf{elif}\;\alpha \le 2.266980258015373608283608969920590572301 \cdot 10^{166}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta + \alpha}{\frac{\left(2 + \left(\beta + \alpha\right)\right) + i \cdot 2}{\frac{\beta - \alpha}{\left(i \cdot 2 + \beta\right) + \alpha}}}\right)}^{3}} + 1}{2}\\ \end{array}\]
\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 9.76982660809475993097637758871956020598 \cdot 10^{62}:\\
\;\;\;\;\frac{1 + \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(i \cdot 2 + \alpha\right) + \beta} \cdot \left(\beta - \alpha\right)\right)}{2 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{2}\\

\mathbf{elif}\;\alpha \le 2.266980258015373608283608969920590572301 \cdot 10^{166}:\\
\;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta + \alpha}{\frac{\left(2 + \left(\beta + \alpha\right)\right) + i \cdot 2}{\frac{\beta - \alpha}{\left(i \cdot 2 + \beta\right) + \alpha}}}\right)}^{3}} + 1}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r108043 = alpha;
        double r108044 = beta;
        double r108045 = r108043 + r108044;
        double r108046 = r108044 - r108043;
        double r108047 = r108045 * r108046;
        double r108048 = 2.0;
        double r108049 = i;
        double r108050 = r108048 * r108049;
        double r108051 = r108045 + r108050;
        double r108052 = r108047 / r108051;
        double r108053 = r108051 + r108048;
        double r108054 = r108052 / r108053;
        double r108055 = 1.0;
        double r108056 = r108054 + r108055;
        double r108057 = r108056 / r108048;
        return r108057;
}
↓
double f(double alpha, double beta, double i) {
        double r108058 = alpha;
        double r108059 = 9.76982660809476e+62;
        bool r108060 = r108058 <= r108059;
        double r108061 = 1.0;
        double r108062 = beta;
        double r108063 = r108062 + r108058;
        double r108064 = 1.0;
        double r108065 = i;
        double r108066 = 2.0;
        double r108067 = r108065 * r108066;
        double r108068 = r108067 + r108058;
        double r108069 = r108068 + r108062;
        double r108070 = r108064 / r108069;
        double r108071 = r108062 - r108058;
        double r108072 = r108070 * r108071;
        double r108073 = r108063 * r108072;
        double r108074 = r108063 + r108067;
        double r108075 = r108066 + r108074;
        double r108076 = r108073 / r108075;
        double r108077 = r108061 + r108076;
        double r108078 = r108077 / r108066;
        double r108079 = 2.2669802580153736e+166;
        bool r108080 = r108058 <= r108079;
        double r108081 = r108066 / r108058;
        double r108082 = 8.0;
        double r108083 = 3.0;
        double r108084 = pow(r108058, r108083);
        double r108085 = r108082 / r108084;
        double r108086 = 4.0;
        double r108087 = r108058 * r108058;
        double r108088 = r108086 / r108087;
        double r108089 = r108085 - r108088;
        double r108090 = r108081 + r108089;
        double r108091 = r108090 / r108066;
        double r108092 = r108066 + r108063;
        double r108093 = r108092 + r108067;
        double r108094 = r108067 + r108062;
        double r108095 = r108094 + r108058;
        double r108096 = r108071 / r108095;
        double r108097 = r108093 / r108096;
        double r108098 = r108063 / r108097;
        double r108099 = pow(r108098, r108083);
        double r108100 = cbrt(r108099);
        double r108101 = r108100 + r108061;
        double r108102 = r108101 / r108066;
        double r108103 = r108080 ? r108091 : r108102;
        double r108104 = r108060 ? r108078 : r108103;
        return r108104;
}

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 3 regimes

  2. if alpha < 9.76982660809476e+62

    1. Initial program 12.1

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

    3. Applied *-un-lft-identity12.1

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

    4. Applied times-frac1.6

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

    5. Simplified1.6

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

    6. Simplified1.6

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

    8. Applied div-inv1.6

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

    9. Simplified1.6

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

    if 9.76982660809476e+62 < alpha < 2.2669802580153736e+166

    1. Initial program 45.7

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

    2. Taylor expanded around inf 39.9

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

    3. Simplified39.9

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

    if 2.2669802580153736e+166 < 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. Using strategy rm

    3. Applied *-un-lft-identity64.0

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

    4. Applied times-frac47.4

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

    5. Simplified47.4

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

    6. Simplified47.4

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

    8. Applied div-inv47.4

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

    9. Simplified47.4

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

    11. Applied add-cbrt-cube53.4

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

    12. Applied add-cbrt-cube53.4

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

    13. Applied add-cbrt-cube64.0

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

    14. Applied cbrt-unprod64.0

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

    15. Applied cbrt-undiv64.0

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

    16. Simplified47.4

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{\alpha + \beta}{\frac{\left(2 + \left(\alpha + \beta\right)\right) + i \cdot 2}{\frac{\beta - \alpha}{\alpha + \left(i \cdot 2 + \beta\right)}}}\right)}^{3}}} + 1}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.76982660809475993097637758871956020598 \cdot 10^{62}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\left(i \cdot 2 + \alpha\right) + \beta} \cdot \left(\beta - \alpha\right)\right)}{2 + \left(\left(\beta + \alpha\right) + i \cdot 2\right)}}{2}\\ \mathbf{elif}\;\alpha \le 2.266980258015373608283608969920590572301 \cdot 10^{166}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{{\alpha}^{3}} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\beta + \alpha}{\frac{\left(2 + \left(\beta + \alpha\right)\right) + i \cdot 2}{\frac{\beta - \alpha}{\left(i \cdot 2 + \beta\right) + \alpha}}}\right)}^{3}} + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))