Octave 3.8, jcobi/4

Average Error: 54.1 → 13.8

Time: 15.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.200334102707676578905198390790180852224 \cdot 10^{64}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}{i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{elif}\;i \le 1.373853224073776070930027931834659728415 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(8, {i}^{2}, \mathsf{fma}\left(12, \alpha \cdot i, 12 \cdot \left(i \cdot \beta\right)\right)\right)}}{\frac{2}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r117706 = i;
        double r117707 = alpha;
        double r117708 = beta;
        double r117709 = r117707 + r117708;
        double r117710 = r117709 + r117706;
        double r117711 = r117706 * r117710;
        double r117712 = r117708 * r117707;
        double r117713 = r117712 + r117711;
        double r117714 = r117711 * r117713;
        double r117715 = 2.0;
        double r117716 = r117715 * r117706;
        double r117717 = r117709 + r117716;
        double r117718 = r117717 * r117717;
        double r117719 = r117714 / r117718;
        double r117720 = 1.0;
        double r117721 = r117718 - r117720;
        double r117722 = r117719 / r117721;
        return r117722;
}

↓

double f(double alpha, double beta, double i) {
        double r117723 = i;
        double r117724 = 1.2003341027076766e+64;
        bool r117725 = r117723 <= r117724;
        double r117726 = alpha;
        double r117727 = beta;
        double r117728 = r117726 + r117727;
        double r117729 = r117728 + r117723;
        double r117730 = 2.0;
        double r117731 = r117730 * r117723;
        double r117732 = r117728 + r117731;
        double r117733 = 3.0;
        double r117734 = pow(r117732, r117733);
        double r117735 = fma(r117723, r117730, r117728);
        double r117736 = 1.0;
        double r117737 = -r117736;
        double r117738 = r117735 * r117737;
        double r117739 = r117734 + r117738;
        double r117740 = r117739 / r117723;
        double r117741 = r117729 / r117740;
        double r117742 = r117723 * r117729;
        double r117743 = fma(r117727, r117726, r117742);
        double r117744 = r117735 / r117743;
        double r117745 = r117741 / r117744;
        double r117746 = 1.373853224073776e+125;
        bool r117747 = r117723 <= r117746;
        double r117748 = 8.0;
        double r117749 = 2.0;
        double r117750 = pow(r117723, r117749);
        double r117751 = 12.0;
        double r117752 = r117726 * r117723;
        double r117753 = r117723 * r117727;
        double r117754 = r117751 * r117753;
        double r117755 = fma(r117751, r117752, r117754);
        double r117756 = fma(r117748, r117750, r117755);
        double r117757 = r117729 / r117756;
        double r117758 = r117730 / r117723;
        double r117759 = r117757 / r117758;
        double r117760 = 0.0625;
        double r117761 = r117747 ? r117759 : r117760;
        double r117762 = r117725 ? r117745 : r117761;
        return r117762;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 1.2003341027076766e+64

   1. Initial program 23.1

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

   2. Simplified 19.3

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

   4. Applied *-un-lft-identity 19.3

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

   5. Applied times-frac 16.2

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

   6. Applied associate-/r* 16.2

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

   7. Simplified 16.2

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

   if 1.2003341027076766e+64 < i < 1.373853224073776e+125

   1. Initial program 57.2

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

   2. Simplified 55.8

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

   4. Applied *-un-lft-identity 55.8

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

   5. Applied times-frac 39.7

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

   6. Applied associate-/r* 39.7

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

   7. Simplified 39.7

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

   8. Taylor expanded around inf 25.5

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

   9. Simplified 25.5

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

   10. Taylor expanded around inf 19.3

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

   if 1.373853224073776e+125 < 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 63.8

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

   4. Applied *-un-lft-identity 63.8

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

   5. Applied times-frac 63.8

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

   6. Applied associate-/r* 63.8

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

   7. Simplified 63.8

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

   8. Taylor expanded around inf 57.4

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

   9. Simplified 57.4

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

   10. Taylor expanded around 0 11.1

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

4. Final simplification 13.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.200334102707676578905198390790180852224 \cdot 10^{64}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{{\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{3} + \mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \left(-1\right)}{i}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}\\ \mathbf{elif}\;i \le 1.373853224073776070930027931834659728415 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(8, {i}^{2}, \mathsf{fma}\left(12, \alpha \cdot i, 12 \cdot \left(i \cdot \beta\right)\right)\right)}}{\frac{2}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))