Octave 3.8, jcobi/2

Average Error: 23.9 → 7.6

Time: 10.7s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}\right)}^{3}}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r117711 = alpha;
        double r117712 = beta;
        double r117713 = r117711 + r117712;
        double r117714 = r117712 - r117711;
        double r117715 = r117713 * r117714;
        double r117716 = 2.0;
        double r117717 = i;
        double r117718 = r117716 * r117717;
        double r117719 = r117713 + r117718;
        double r117720 = r117715 / r117719;
        double r117721 = r117719 + r117716;
        double r117722 = r117720 / r117721;
        double r117723 = 1.0;
        double r117724 = r117722 + r117723;
        double r117725 = r117724 / r117716;
        return r117725;
}

↓

double f(double alpha, double beta, double i) {
        double r117726 = alpha;
        double r117727 = beta;
        double r117728 = r117726 + r117727;
        double r117729 = r117727 - r117726;
        double r117730 = r117728 * r117729;
        double r117731 = 2.0;
        double r117732 = i;
        double r117733 = r117731 * r117732;
        double r117734 = r117728 + r117733;
        double r117735 = r117730 / r117734;
        double r117736 = r117734 + r117731;
        double r117737 = r117735 / r117736;
        double r117738 = -1.0;
        bool r117739 = r117737 <= r117738;
        double r117740 = 1.0;
        double r117741 = r117740 / r117726;
        double r117742 = 8.0;
        double r117743 = 3.0;
        double r117744 = pow(r117726, r117743);
        double r117745 = r117740 / r117744;
        double r117746 = r117742 * r117745;
        double r117747 = 4.0;
        double r117748 = 2.0;
        double r117749 = pow(r117726, r117748);
        double r117750 = r117740 / r117749;
        double r117751 = r117747 * r117750;
        double r117752 = r117746 - r117751;
        double r117753 = fma(r117731, r117741, r117752);
        double r117754 = r117753 / r117731;
        double r117755 = r117729 / r117734;
        double r117756 = r117755 / r117736;
        double r117757 = 1.0;
        double r117758 = fma(r117728, r117756, r117757);
        double r117759 = pow(r117758, r117743);
        double r117760 = cbrt(r117759);
        double r117761 = pow(r117760, r117743);
        double r117762 = cbrt(r117761);
        double r117763 = pow(r117762, r117743);
        double r117764 = cbrt(r117763);
        double r117765 = r117764 / r117731;
        double r117766 = r117739 ? r117754 : r117765;
        return r117766;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) < -1.0

   1. Initial program 63.3

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

   2. Taylor expanded around inf 32.9

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 32.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2}\]

   if -1.0 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))

   1. Initial program 12.9

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

   3. Applied *-un-lft-identity 12.9

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

   4. Applied *-un-lft-identity 12.9

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

   5. Applied times-frac 0.5

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

   6. Applied times-frac 0.5

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

   7. Applied fma-def 0.5

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

   9. Applied add-cbrt-cube 0.5

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

   10. Simplified 0.5

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

   12. Applied add-cbrt-cube 0.5

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

   13. Simplified 0.5

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

   15. Applied add-cbrt-cube 0.5

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

   16. Simplified 0.5

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -1:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))