Average Error: 24.0 → 11.4
Time: 4.0m
Precision: 64
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.1178533449480407 \cdot 10^{+147}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{elif}\;\alpha \le 3.592412999901975 \cdot 10^{+171}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\alpha \le 1.2439718835440208 \cdot 10^{+232}:\\ \;\;\;\;\frac{(\left(\frac{\beta + \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]
double f(double alpha, double beta, double i) {
        double r15263561 = alpha;
        double r15263562 = beta;
        double r15263563 = r15263561 + r15263562;
        double r15263564 = r15263562 - r15263561;
        double r15263565 = r15263563 * r15263564;
        double r15263566 = 2.0;
        double r15263567 = i;
        double r15263568 = r15263566 * r15263567;
        double r15263569 = r15263563 + r15263568;
        double r15263570 = r15263565 / r15263569;
        double r15263571 = 2.0;
        double r15263572 = r15263569 + r15263571;
        double r15263573 = r15263570 / r15263572;
        double r15263574 = 1.0;
        double r15263575 = r15263573 + r15263574;
        double r15263576 = r15263575 / r15263571;
        return r15263576;
}
↓
double f(double alpha, double beta, double i) {
        double r15263577 = alpha;
        double r15263578 = 1.1178533449480407e+147;
        bool r15263579 = r15263577 <= r15263578;
        double r15263580 = beta;
        double r15263581 = r15263580 + r15263577;
        double r15263582 = r15263580 - r15263577;
        double r15263583 = 2.0;
        double r15263584 = i;
        double r15263585 = r15263583 * r15263584;
        double r15263586 = r15263581 + r15263585;
        double r15263587 = r15263582 / r15263586;
        double r15263588 = 2.0;
        double r15263589 = r15263588 + r15263586;
        double r15263590 = r15263587 / r15263589;
        double r15263591 = 1.0;
        double r15263592 = fma(r15263581, r15263590, r15263591);
        double r15263593 = r15263592 / r15263588;
        double r15263594 = 3.592412999901975e+171;
        bool r15263595 = r15263577 <= r15263594;
        double r15263596 = 1.0;
        double r15263597 = r15263577 * r15263577;
        double r15263598 = r15263596 / r15263597;
        double r15263599 = 8.0;
        double r15263600 = r15263599 / r15263577;
        double r15263601 = 4.0;
        double r15263602 = r15263600 - r15263601;
        double r15263603 = r15263588 / r15263577;
        double r15263604 = fma(r15263598, r15263602, r15263603);
        double r15263605 = r15263604 / r15263588;
        double r15263606 = 1.2439718835440208e+232;
        bool r15263607 = r15263577 <= r15263606;
        double r15263608 = sqrt(r15263586);
        double r15263609 = r15263581 / r15263608;
        double r15263610 = r15263582 / r15263608;
        double r15263611 = r15263610 / r15263589;
        double r15263612 = fma(r15263609, r15263611, r15263591);
        double r15263613 = r15263612 / r15263588;
        double r15263614 = r15263607 ? r15263613 : r15263605;
        double r15263615 = r15263595 ? r15263605 : r15263614;
        double r15263616 = r15263579 ? r15263593 : r15263615;
        return r15263616;
}


\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 1.1178533449480407 \cdot 10^{+147}:\\
\;\;\;\;\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 3.592412999901975 \cdot 10^{+171}:\\
\;\;\;\;\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 1.2439718835440208 \cdot 10^{+232}:\\
\;\;\;\;\frac{(\left(\frac{\beta + \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\sqrt{\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}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if alpha < 1.1178533449480407e+147

    1. Initial program 16.0

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

    3. Applied *-un-lft-identity16.0

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

    4. Applied *-un-lft-identity16.0

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

    5. Applied distribute-lft-out16.0

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

    6. Applied *-un-lft-identity16.0

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

    7. Applied times-frac5.2

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

    8. Applied times-frac5.2

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

    9. Applied fma-def5.2

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

    10. 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 1.1178533449480407e+147 < alpha < 3.592412999901975e+171 or 1.2439718835440208e+232 < alpha

    1. Initial program 62.3

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

    3. Applied add-sqr-sqrt62.3

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

    4. Applied *-un-lft-identity62.3

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

    5. Applied times-frac49.4

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

    6. Applied times-frac49.4

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

    7. Taylor expanded around inf 40.6

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

    8. Simplified40.6

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

    if 3.592412999901975e+171 < alpha < 1.2439718835440208e+232

    1. Initial program 63.2

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

    3. Applied *-un-lft-identity63.2

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

    4. Applied add-sqr-sqrt63.2

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

    5. Applied times-frac42.8

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

    6. Applied times-frac42.8

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

    7. Applied fma-def43.0

      \[\leadsto \frac{\color{blue}{(\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{1}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0)_*}}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.1178533449480407 \cdot 10^{+147}:\\ \;\;\;\;\frac{(\left(\beta + \alpha\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{elif}\;\alpha \le 3.592412999901975 \cdot 10^{+171}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\alpha \le 1.2439718835440208 \cdot 10^{+232}:\\ \;\;\;\;\frac{(\left(\frac{\beta + \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}}\right) \cdot \left(\frac{\frac{\beta - \alpha}{\sqrt{\left(\beta + \alpha\right) + 2 \cdot i}}}{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}\right) + 1.0)_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +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))