Average Error: 60.3 → 54.6
Time: 36.3s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.108488603526266601185065312427003011215 \cdot 10^{235}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\\ \mathbf{elif}\;b \le -1.428922980252274523091327601335334813994 \cdot 10^{68}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;b \le 1.350733200853811175136047853945361564279 \cdot 10^{52}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\left(\varepsilon \cdot b\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right) \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}}\\ \end{array}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.108488603526266601185065312427003011215 \cdot 10^{235}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\\

\mathbf{elif}\;b \le -1.428922980252274523091327601335334813994 \cdot 10^{68}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\

\mathbf{elif}\;b \le 1.350733200853811175136047853945361564279 \cdot 10^{52}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\left(\varepsilon \cdot b\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right) \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}}\\

\end{array}
double f(double a, double b, double eps) {
        double r78827 = eps;
        double r78828 = a;
        double r78829 = b;
        double r78830 = r78828 + r78829;
        double r78831 = r78830 * r78827;
        double r78832 = exp(r78831);
        double r78833 = 1.0;
        double r78834 = r78832 - r78833;
        double r78835 = r78827 * r78834;
        double r78836 = r78828 * r78827;
        double r78837 = exp(r78836);
        double r78838 = r78837 - r78833;
        double r78839 = r78829 * r78827;
        double r78840 = exp(r78839);
        double r78841 = r78840 - r78833;
        double r78842 = r78838 * r78841;
        double r78843 = r78835 / r78842;
        return r78843;
}
↓
double f(double a, double b, double eps) {
        double r78844 = b;
        double r78845 = -3.1084886035262666e+235;
        bool r78846 = r78844 <= r78845;
        double r78847 = eps;
        double r78848 = a;
        double r78849 = r78848 + r78844;
        double r78850 = r78849 * r78847;
        double r78851 = exp(r78850);
        double r78852 = 1.0;
        double r78853 = r78851 - r78852;
        double r78854 = r78847 * r78853;
        double r78855 = r78848 * r78847;
        double r78856 = exp(r78855);
        double r78857 = r78856 - r78852;
        double r78858 = r78844 * r78847;
        double r78859 = exp(r78858);
        double r78860 = r78859 - r78852;
        double r78861 = r78857 * r78860;
        double r78862 = exp(r78861);
        double r78863 = log(r78862);
        double r78864 = r78854 / r78863;
        double r78865 = -1.4289229802522745e+68;
        bool r78866 = r78844 <= r78865;
        double r78867 = 0.16666666666666666;
        double r78868 = 3.0;
        double r78869 = pow(r78848, r78868);
        double r78870 = pow(r78847, r78868);
        double r78871 = r78869 * r78870;
        double r78872 = 0.5;
        double r78873 = 2.0;
        double r78874 = pow(r78848, r78873);
        double r78875 = pow(r78847, r78873);
        double r78876 = r78874 * r78875;
        double r78877 = fma(r78872, r78876, r78855);
        double r78878 = fma(r78867, r78871, r78877);
        double r78879 = r78878 * r78860;
        double r78880 = r78854 / r78879;
        double r78881 = 1.3507332008538112e+52;
        bool r78882 = r78844 <= r78881;
        double r78883 = r78847 * r78844;
        double r78884 = pow(r78883, r78868);
        double r78885 = pow(r78844, r78873);
        double r78886 = r78875 * r78885;
        double r78887 = fma(r78872, r78886, r78858);
        double r78888 = fma(r78867, r78884, r78887);
        double r78889 = r78857 * r78888;
        double r78890 = r78854 / r78889;
        double r78891 = cbrt(r78861);
        double r78892 = r78891 * r78891;
        double r78893 = r78892 * r78891;
        double r78894 = r78854 / r78893;
        double r78895 = r78882 ? r78890 : r78894;
        double r78896 = r78866 ? r78880 : r78895;
        double r78897 = r78846 ? r78864 : r78896;
        return r78897;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original60.3
Target14.9
Herbie54.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.1084886035262666e+235

    1. Initial program 46.7

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp47.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\log \left(e^{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}}\]

    if -3.1084886035262666e+235 < b < -1.4289229802522745e+68

    1. Initial program 55.6

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    2. Taylor expanded around 0 49.4

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    3. Simplified49.4

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    if -1.4289229802522745e+68 < b < 1.3507332008538112e+52

    1. Initial program 63.5

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

    2. Taylor expanded around 0 56.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]

    3. Simplified56.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}}\]
    4. Using strategy rm

    5. Applied pow-prod-down56.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{\left(\varepsilon \cdot b\right)}^{3}}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\]

    if 1.3507332008538112e+52 < b

    1. Initial program 54.3

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt54.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right) \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification54.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.108488603526266601185065312427003011215 \cdot 10^{235}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\log \left(e^{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\\ \mathbf{elif}\;b \le -1.428922980252274523091327601335334813994 \cdot 10^{68}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{elif}\;b \le 1.350733200853811175136047853945361564279 \cdot 10^{52}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\left(\varepsilon \cdot b\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, b \cdot \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right) \cdot \sqrt[3]{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1 eps) (< eps 1))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))