Average Error: 52.5 → 35.3
Time: 3.2m
Precision: 64
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.8437334604126914 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*}} \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_* - \sqrt{1.0}\right)} \cdot \frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\beta + \alpha\right))_*}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
double f(double alpha, double beta, double i) {
        double r16692568 = i;
        double r16692569 = alpha;
        double r16692570 = beta;
        double r16692571 = r16692569 + r16692570;
        double r16692572 = r16692571 + r16692568;
        double r16692573 = r16692568 * r16692572;
        double r16692574 = r16692570 * r16692569;
        double r16692575 = r16692574 + r16692573;
        double r16692576 = r16692573 * r16692575;
        double r16692577 = 2.0;
        double r16692578 = r16692577 * r16692568;
        double r16692579 = r16692571 + r16692578;
        double r16692580 = r16692579 * r16692579;
        double r16692581 = r16692576 / r16692580;
        double r16692582 = 1.0;
        double r16692583 = r16692580 - r16692582;
        double r16692584 = r16692581 / r16692583;
        return r16692584;
}
↓
double f(double alpha, double beta, double i) {
        double r16692585 = alpha;
        double r16692586 = 3.8437334604126914e+212;
        bool r16692587 = r16692585 <= r16692586;
        double r16692588 = 1.0;
        double r16692589 = i;
        double r16692590 = beta;
        double r16692591 = r16692590 + r16692585;
        double r16692592 = r16692589 + r16692591;
        double r16692593 = r16692589 * r16692592;
        double r16692594 = 2.0;
        double r16692595 = fma(r16692594, r16692589, r16692591);
        double r16692596 = r16692593 / r16692595;
        double r16692597 = r16692588 / r16692596;
        double r16692598 = 1.0;
        double r16692599 = sqrt(r16692598);
        double r16692600 = r16692595 - r16692599;
        double r16692601 = r16692597 * r16692600;
        double r16692602 = r16692588 / r16692601;
        double r16692603 = r16692590 * r16692585;
        double r16692604 = fma(r16692592, r16692589, r16692603);
        double r16692605 = r16692604 / r16692595;
        double r16692606 = r16692599 + r16692595;
        double r16692607 = r16692605 / r16692606;
        double r16692608 = r16692602 * r16692607;
        double r16692609 = 0.0;
        double r16692610 = r16692587 ? r16692608 : r16692609;
        return r16692610;
}


\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.8437334604126914 \cdot 10^{+212}:\\
\;\;\;\;\frac{1}{\frac{1}{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*}} \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_* - \sqrt{1.0}\right)} \cdot \frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\beta + \alpha\right))_*}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.8437334604126914e+212

    1. Initial program 51.3

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

    2. Simplified51.3

      \[\leadsto \color{blue}{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_* - 1.0}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt51.3

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

    5. Applied difference-of-squares51.3

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

    6. Applied times-frac36.7

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

    7. Applied times-frac34.3

      \[\leadsto \color{blue}{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\]
    8. Using strategy rm

    9. Applied clear-num34.3

      \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \color{blue}{\frac{1}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}}}\]
    10. Using strategy rm

    11. Applied div-inv34.4

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

    if 3.8437334604126914e+212 < alpha

    1. Initial program 62.6

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

    2. Simplified62.6

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

    3. Taylor expanded around -inf 42.6

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification35.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.8437334604126914 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*}} \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_* - \sqrt{1.0}\right)} \cdot \frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\beta + \alpha\right))_*}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))