Average Error: 54.0 → 38.5
Time: 10.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \le 2.134998786884618 \cdot 10^{147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(2 \cdot i\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right) + \left(\alpha + \beta\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;\beta \le 2.134998786884618 \cdot 10^{147}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(2 \cdot i\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right) + \left(\alpha + \beta\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r79622 = i;
        double r79623 = alpha;
        double r79624 = beta;
        double r79625 = r79623 + r79624;
        double r79626 = r79625 + r79622;
        double r79627 = r79622 * r79626;
        double r79628 = r79624 * r79623;
        double r79629 = r79628 + r79627;
        double r79630 = r79627 * r79629;
        double r79631 = 2.0;
        double r79632 = r79631 * r79622;
        double r79633 = r79625 + r79632;
        double r79634 = r79633 * r79633;
        double r79635 = r79630 / r79634;
        double r79636 = 1.0;
        double r79637 = r79634 - r79636;
        double r79638 = r79635 / r79637;
        return r79638;
}
↓
double f(double alpha, double beta, double i) {
        double r79639 = beta;
        double r79640 = 2.134998786884618e+147;
        bool r79641 = r79639 <= r79640;
        double r79642 = alpha;
        double r79643 = i;
        double r79644 = r79642 + r79639;
        double r79645 = r79644 + r79643;
        double r79646 = r79643 * r79645;
        double r79647 = fma(r79639, r79642, r79646);
        double r79648 = 2.0;
        double r79649 = r79648 * r79643;
        double r79650 = fma(r79648, r79643, r79644);
        double r79651 = r79649 * r79650;
        double r79652 = r79644 * r79650;
        double r79653 = r79651 + r79652;
        double r79654 = r79647 / r79653;
        double r79655 = 1.0;
        double r79656 = -r79655;
        double r79657 = fma(r79650, r79650, r79656);
        double r79658 = r79646 / r79657;
        double r79659 = r79654 * r79658;
        double r79660 = 0.0;
        double r79661 = r79641 ? r79659 : r79660;
        return r79661;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.134998786884618e+147

    1. Initial program 51.9

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

    2. Simplified53.1

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

    4. Applied times-frac36.2

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

    6. Applied fma-udef36.2

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

    7. Applied distribute-lft-in36.2

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

    8. Simplified36.2

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

    9. Simplified36.2

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

    if 2.134998786884618e+147 < beta

    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. Simplified64.0

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

    3. Taylor expanded around inf 49.6

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

  4. Final simplification38.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 2.134998786884618 \cdot 10^{147}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(2 \cdot i\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right) + \left(\alpha + \beta\right) \cdot \mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \alpha + \beta\right), \mathsf{fma}\left(2, i, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))