Average Error: 24.0 → 11.4
Time: 21.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.19509086812250328832562519981486685202 \cdot 10^{103}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}{\beta - \alpha}}, \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]
\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}\;\alpha \le 4.19509086812250328832562519981486685202 \cdot 10^{103}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}{\beta - \alpha}}, \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r152893 = alpha;
        double r152894 = beta;
        double r152895 = r152893 + r152894;
        double r152896 = r152894 - r152893;
        double r152897 = r152895 * r152896;
        double r152898 = 2.0;
        double r152899 = i;
        double r152900 = r152898 * r152899;
        double r152901 = r152895 + r152900;
        double r152902 = r152897 / r152901;
        double r152903 = r152901 + r152898;
        double r152904 = r152902 / r152903;
        double r152905 = 1.0;
        double r152906 = r152904 + r152905;
        double r152907 = r152906 / r152898;
        return r152907;
}
↓
double f(double alpha, double beta, double i) {
        double r152908 = alpha;
        double r152909 = 4.195090868122503e+103;
        bool r152910 = r152908 <= r152909;
        double r152911 = 1.0;
        double r152912 = i;
        double r152913 = 2.0;
        double r152914 = beta;
        double r152915 = r152908 + r152914;
        double r152916 = fma(r152912, r152913, r152915);
        double r152917 = r152916 / r152915;
        double r152918 = r152914 - r152908;
        double r152919 = r152917 / r152918;
        double r152920 = r152911 / r152919;
        double r152921 = r152913 * r152912;
        double r152922 = r152915 + r152921;
        double r152923 = r152922 + r152913;
        double r152924 = r152911 / r152923;
        double r152925 = 1.0;
        double r152926 = fma(r152920, r152924, r152925);
        double r152927 = 3.0;
        double r152928 = pow(r152926, r152927);
        double r152929 = cbrt(r152928);
        double r152930 = pow(r152929, r152927);
        double r152931 = cbrt(r152930);
        double r152932 = r152931 / r152913;
        double r152933 = r152911 / r152908;
        double r152934 = 8.0;
        double r152935 = pow(r152908, r152927);
        double r152936 = r152911 / r152935;
        double r152937 = r152934 * r152936;
        double r152938 = 4.0;
        double r152939 = 2.0;
        double r152940 = pow(r152908, r152939);
        double r152941 = r152911 / r152940;
        double r152942 = r152938 * r152941;
        double r152943 = r152937 - r152942;
        double r152944 = fma(r152913, r152933, r152943);
        double r152945 = r152944 / r152913;
        double r152946 = r152910 ? r152932 : r152945;
        return r152946;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.195090868122503e+103

    1. Initial program 13.8

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

    3. Applied clear-num13.8

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

    4. Simplified3.2

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

    6. Applied div-inv3.2

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

    7. Applied fma-def3.2

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

    9. Applied add-cbrt-cube3.2

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

    10. Simplified3.2

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

    12. Applied add-cbrt-cube3.2

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

    13. Simplified3.2

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

    if 4.195090868122503e+103 < alpha

    1. Initial program 59.3

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

    2. Taylor expanded around inf 39.6

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

    3. Simplified39.6

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.19509086812250328832562519981486685202 \cdot 10^{103}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\sqrt[3]{{\left(\mathsf{fma}\left(\frac{1}{\frac{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \beta}}{\beta - \alpha}}, \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)\right)}^{3}}\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{1}{\alpha}, 8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))