Average Error: 52.7 → 36.6
Time: 3.2m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.376827553053855 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\beta + \alpha\right))_*} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{(2 \cdot i + \left(\beta + \alpha\right))_* - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 6.376827553053855 \cdot 10^{+152}:\\
\;\;\;\;\frac{\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))_*} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{(2 \cdot i + \left(\beta + \alpha\right))_* - \sqrt{1.0}}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r18588720 = i;
        double r18588721 = alpha;
        double r18588722 = beta;
        double r18588723 = r18588721 + r18588722;
        double r18588724 = r18588723 + r18588720;
        double r18588725 = r18588720 * r18588724;
        double r18588726 = r18588722 * r18588721;
        double r18588727 = r18588726 + r18588725;
        double r18588728 = r18588725 * r18588727;
        double r18588729 = 2.0;
        double r18588730 = r18588729 * r18588720;
        double r18588731 = r18588723 + r18588730;
        double r18588732 = r18588731 * r18588731;
        double r18588733 = r18588728 / r18588732;
        double r18588734 = 1.0;
        double r18588735 = r18588732 - r18588734;
        double r18588736 = r18588733 / r18588735;
        return r18588736;
}
↓
double f(double alpha, double beta, double i) {
        double r18588737 = alpha;
        double r18588738 = 6.376827553053855e+152;
        bool r18588739 = r18588737 <= r18588738;
        double r18588740 = i;
        double r18588741 = beta;
        double r18588742 = r18588741 + r18588737;
        double r18588743 = r18588740 + r18588742;
        double r18588744 = r18588741 * r18588737;
        double r18588745 = fma(r18588743, r18588740, r18588744);
        double r18588746 = 2.0;
        double r18588747 = fma(r18588746, r18588740, r18588742);
        double r18588748 = r18588745 / r18588747;
        double r18588749 = 1.0;
        double r18588750 = sqrt(r18588749);
        double r18588751 = r18588750 + r18588747;
        double r18588752 = r18588748 / r18588751;
        double r18588753 = r18588740 * r18588743;
        double r18588754 = r18588753 / r18588747;
        double r18588755 = r18588752 * r18588754;
        double r18588756 = r18588747 - r18588750;
        double r18588757 = r18588755 / r18588756;
        double r18588758 = 0.0;
        double r18588759 = r18588739 ? r18588757 : r18588758;
        return r18588759;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6.376827553053855e+152

    1. Initial program 50.7

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

    2. Simplified50.7

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

    4. Applied add-sqr-sqrt50.7

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

    5. Applied difference-of-squares50.7

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

    6. Applied times-frac35.6

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

    7. Applied times-frac34.4

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

    9. Applied associate-*r/34.4

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

    if 6.376827553053855e+152 < alpha

    1. Initial program 62.5

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

    2. Simplified62.5

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

    4. Applied add-sqr-sqrt62.5

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

    5. Applied difference-of-squares62.5

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

    6. Applied times-frac56.0

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

    7. Applied times-frac49.2

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

    8. Taylor expanded around inf 47.7

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

  4. Final simplification36.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.376827553053855 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{\frac{(\left(i + \left(\beta + \alpha\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\beta + \alpha\right))_*} \cdot \frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{(2 \cdot i + \left(\beta + \alpha\right))_* - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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)))