Average Error: 23.6 → 11.8
Time: 5.6m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.097583333065985 \cdot 10^{+102}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{elif}\;\alpha \le 2.1783246613359708 \cdot 10^{+131}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\alpha \le 1.1393860257411403 \cdot 10^{+174}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 2.097583333065985 \cdot 10^{+102}:\\
\;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\

\mathbf{elif}\;\alpha \le 2.1783246613359708 \cdot 10^{+131}:\\
\;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\

\mathbf{elif}\;\alpha \le 1.1393860257411403 \cdot 10^{+174}:\\
\;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r24660527 = alpha;
        double r24660528 = beta;
        double r24660529 = r24660527 + r24660528;
        double r24660530 = r24660528 - r24660527;
        double r24660531 = r24660529 * r24660530;
        double r24660532 = 2.0;
        double r24660533 = i;
        double r24660534 = r24660532 * r24660533;
        double r24660535 = r24660529 + r24660534;
        double r24660536 = r24660531 / r24660535;
        double r24660537 = 2.0;
        double r24660538 = r24660535 + r24660537;
        double r24660539 = r24660536 / r24660538;
        double r24660540 = 1.0;
        double r24660541 = r24660539 + r24660540;
        double r24660542 = r24660541 / r24660537;
        return r24660542;
}
↓
double f(double alpha, double beta, double i) {
        double r24660543 = alpha;
        double r24660544 = 2.097583333065985e+102;
        bool r24660545 = r24660543 <= r24660544;
        double r24660546 = beta;
        double r24660547 = r24660546 + r24660543;
        double r24660548 = r24660546 - r24660543;
        double r24660549 = 2.0;
        double r24660550 = i;
        double r24660551 = r24660549 * r24660550;
        double r24660552 = r24660547 + r24660551;
        double r24660553 = r24660548 / r24660552;
        double r24660554 = 2.0;
        double r24660555 = r24660554 + r24660552;
        double r24660556 = r24660553 / r24660555;
        double r24660557 = 1.0;
        double r24660558 = fma(r24660547, r24660556, r24660557);
        double r24660559 = r24660558 / r24660554;
        double r24660560 = 2.1783246613359708e+131;
        bool r24660561 = r24660543 <= r24660560;
        double r24660562 = 1.0;
        double r24660563 = r24660543 * r24660543;
        double r24660564 = r24660562 / r24660563;
        double r24660565 = 8.0;
        double r24660566 = r24660565 / r24660543;
        double r24660567 = 4.0;
        double r24660568 = r24660566 - r24660567;
        double r24660569 = r24660554 / r24660543;
        double r24660570 = fma(r24660564, r24660568, r24660569);
        double r24660571 = r24660570 / r24660554;
        double r24660572 = 1.1393860257411403e+174;
        bool r24660573 = r24660543 <= r24660572;
        double r24660574 = r24660573 ? r24660559 : r24660571;
        double r24660575 = r24660561 ? r24660571 : r24660574;
        double r24660576 = r24660545 ? r24660559 : r24660575;
        return r24660576;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.097583333065985e+102 or 2.1783246613359708e+131 < alpha < 1.1393860257411403e+174

    1. Initial program 15.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity15.7

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity15.7

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    5. Applied times-frac5.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    6. Applied times-frac5.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    7. Applied fma-def5.2

      \[\leadsto \frac{\color{blue}{(\left(\frac{\frac{\alpha + \beta}{1}}{1}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0)_*}}{2.0}\]

    8. Simplified5.2

      \[\leadsto \frac{(\color{blue}{\left(\beta + \alpha\right)} \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0)_*}{2.0}\]

    if 2.097583333065985e+102 < alpha < 2.1783246613359708e+131 or 1.1393860257411403e+174 < alpha

    1. Initial program 59.6

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied associate-/l*45.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    4. Taylor expanded around inf 41.8

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    5. Simplified41.8

      \[\leadsto \frac{\color{blue}{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.097583333065985 \cdot 10^{+102}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{elif}\;\alpha \le 2.1783246613359708 \cdot 10^{+131}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\alpha \le 1.1393860257411403 \cdot 10^{+174}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))