Octave 3.8, jcobi/4

Average Error: 54.3 → 41.6

Time: 14.0s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 6.388557784475248968732590130499879730187 \cdot 10^{101}:\\ \;\;\;\;\frac{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), -1, {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{3}\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\mathsf{fma}\left(3, \alpha, \mathsf{fma}\left(6, i, 3 \cdot \beta\right)\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r114613 = i;
        double r114614 = alpha;
        double r114615 = beta;
        double r114616 = r114614 + r114615;
        double r114617 = r114616 + r114613;
        double r114618 = r114613 * r114617;
        double r114619 = r114615 * r114614;
        double r114620 = r114619 + r114618;
        double r114621 = r114618 * r114620;
        double r114622 = 2.0;
        double r114623 = r114622 * r114613;
        double r114624 = r114616 + r114623;
        double r114625 = r114624 * r114624;
        double r114626 = r114621 / r114625;
        double r114627 = 1.0;
        double r114628 = r114625 - r114627;
        double r114629 = r114626 / r114628;
        return r114629;
}

↓

double f(double alpha, double beta, double i) {
        double r114630 = i;
        double r114631 = 6.388557784475249e+101;
        bool r114632 = r114630 <= r114631;
        double r114633 = alpha;
        double r114634 = beta;
        double r114635 = r114633 + r114634;
        double r114636 = r114635 + r114630;
        double r114637 = 1.0;
        double r114638 = 2.0;
        double r114639 = fma(r114630, r114638, r114635);
        double r114640 = r114637 / r114639;
        double r114641 = r114636 * r114640;
        double r114642 = r114630 * r114641;
        double r114643 = 1.0;
        double r114644 = -r114643;
        double r114645 = 3.0;
        double r114646 = pow(r114639, r114645);
        double r114647 = fma(r114639, r114644, r114646);
        double r114648 = r114630 * r114636;
        double r114649 = fma(r114634, r114633, r114648);
        double r114650 = r114647 / r114649;
        double r114651 = r114642 / r114650;
        double r114652 = 6.0;
        double r114653 = r114645 * r114634;
        double r114654 = fma(r114652, r114630, r114653);
        double r114655 = fma(r114645, r114633, r114654);
        double r114656 = r114642 / r114655;
        double r114657 = r114632 ? r114651 : r114656;
        return r114657;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 6.388557784475249e+101

   1. Initial program 33.9

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

   2. Simplified 29.7

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

   4. Applied div-inv 29.7

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

   5. Applied associate-/r* 29.7

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

   6. Simplified 19.0

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

   8. Applied div-inv 19.1

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

   9. Applied associate-/l* 19.1

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

   10. Simplified 19.0

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

   12. Applied div-inv 19.1

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

   if 6.388557784475249e+101 < 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. Simplified 63.6

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

   4. Applied div-inv 63.6

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

   5. Applied associate-/r* 63.6

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

   6. Simplified 63.4

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

   8. Applied div-inv 63.4

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

   9. Applied associate-/l* 63.4

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

   10. Simplified 63.4

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

   12. Applied div-inv 63.4

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

   13. Taylor expanded around 0 52.3

       \[\leadsto \frac{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\color{blue}{3 \cdot \alpha + \left(6 \cdot i + 3 \cdot \beta\right)}}\]

   14. Simplified 52.3

       \[\leadsto \frac{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\color{blue}{\mathsf{fma}\left(3, \alpha, \mathsf{fma}\left(6, i, 3 \cdot \beta\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.388557784475248968732590130499879730187 \cdot 10^{101}:\\ \;\;\;\;\frac{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), -1, {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{3}\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\mathsf{fma}\left(3, \alpha, \mathsf{fma}\left(6, i, 3 \cdot \beta\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))