Average Error: 54.1 → 15.1
Time: 14.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1.10595476891289156 \cdot 10^{89}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}{i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;i \le 1.10595476891289156 \cdot 10^{89}:\\
\;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}{i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;0.0625\\

\end{array}
double f(double alpha, double beta, double i) {
        double r114733 = i;
        double r114734 = alpha;
        double r114735 = beta;
        double r114736 = r114734 + r114735;
        double r114737 = r114736 + r114733;
        double r114738 = r114733 * r114737;
        double r114739 = r114735 * r114734;
        double r114740 = r114739 + r114738;
        double r114741 = r114738 * r114740;
        double r114742 = 2.0;
        double r114743 = r114742 * r114733;
        double r114744 = r114736 + r114743;
        double r114745 = r114744 * r114744;
        double r114746 = r114741 / r114745;
        double r114747 = 1.0;
        double r114748 = r114745 - r114747;
        double r114749 = r114746 / r114748;
        return r114749;
}
↓
double f(double alpha, double beta, double i) {
        double r114750 = i;
        double r114751 = 1.1059547689128916e+89;
        bool r114752 = r114750 <= r114751;
        double r114753 = alpha;
        double r114754 = beta;
        double r114755 = r114753 + r114754;
        double r114756 = r114755 + r114750;
        double r114757 = 2.0;
        double r114758 = r114757 * r114750;
        double r114759 = r114755 + r114758;
        double r114760 = 3.0;
        double r114761 = pow(r114759, r114760);
        double r114762 = fma(r114750, r114757, r114755);
        double r114763 = 1.0;
        double r114764 = -r114763;
        double r114765 = r114762 * r114764;
        double r114766 = r114761 + r114765;
        double r114767 = r114766 / r114750;
        double r114768 = r114756 / r114767;
        double r114769 = r114750 * r114756;
        double r114770 = fma(r114754, r114753, r114769);
        double r114771 = r114762 / r114770;
        double r114772 = r114768 / r114771;
        double r114773 = 0.0625;
        double r114774 = r114752 ? r114772 : r114773;
        return r114774;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 1.1059547689128916e+89

    1. Initial program 29.8

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

    2. Simplified26.4

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

    4. Applied *-un-lft-identity26.4

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

    5. Applied times-frac19.1

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

    6. Applied associate-/r*19.0

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

    7. Simplified19.0

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

    if 1.1059547689128916e+89 < i

    1. Initial program 64.0

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

    2. Simplified63.5

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

    4. Applied *-un-lft-identity63.5

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

    5. Applied times-frac61.2

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

    6. Applied associate-/r*61.2

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

    7. Simplified61.2

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

    8. Taylor expanded around inf 51.8

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + i}{\color{blue}{8 \cdot {i}^{2} + \left(12 \cdot \left(\alpha \cdot i\right) + 12 \cdot \left(i \cdot \beta\right)\right)}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    9. Simplified51.8

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + i}{\color{blue}{\mathsf{fma}\left(8, {i}^{2}, \mathsf{fma}\left(12, \alpha \cdot i, 12 \cdot \left(i \cdot \beta\right)\right)\right)}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\]

    10. Taylor expanded around inf 49.2

      \[\leadsto \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(8, {i}^{2}, \mathsf{fma}\left(12, \alpha \cdot i, 12 \cdot \left(i \cdot \beta\right)\right)\right)}}{\color{blue}{\frac{2}{i}}}\]

    11. Taylor expanded around 0 13.5

      \[\leadsto \color{blue}{0.0625}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.10595476891289156 \cdot 10^{89}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}{i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))