Average Error: 60.3 → 53.3
Time: 13.4s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1403574947999459 \cdot 10^{116} \lor \neg \left(b \le 7.604732750432952 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot \left(\left({a}^{2} \cdot {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{e^{b \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \end{array}\]
\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 -2.1403574947999459 \cdot 10^{116} \lor \neg \left(b \le 7.604732750432952 \cdot 10^{-35}\right):\\
\;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot \left(\left({a}^{2} \cdot {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{e^{b \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}\\

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

\end{array}
double f(double a, double b, double eps) {
        double r117031 = eps;
        double r117032 = a;
        double r117033 = b;
        double r117034 = r117032 + r117033;
        double r117035 = r117034 * r117031;
        double r117036 = exp(r117035);
        double r117037 = 1.0;
        double r117038 = r117036 - r117037;
        double r117039 = r117031 * r117038;
        double r117040 = r117032 * r117031;
        double r117041 = exp(r117040);
        double r117042 = r117041 - r117037;
        double r117043 = r117033 * r117031;
        double r117044 = exp(r117043);
        double r117045 = r117044 - r117037;
        double r117046 = r117042 * r117045;
        double r117047 = r117039 / r117046;
        return r117047;
}
↓
double f(double a, double b, double eps) {
        double r117048 = b;
        double r117049 = -2.140357494799946e+116;
        bool r117050 = r117048 <= r117049;
        double r117051 = 7.604732750432952e-35;
        bool r117052 = r117048 <= r117051;
        double r117053 = !r117052;
        bool r117054 = r117050 || r117053;
        double r117055 = eps;
        double r117056 = a;
        double r117057 = r117056 + r117048;
        double r117058 = r117057 * r117055;
        double r117059 = exp(r117058);
        double r117060 = 1.0;
        double r117061 = r117059 - r117060;
        double r117062 = r117055 * r117061;
        double r117063 = 0.16666666666666666;
        double r117064 = r117055 * r117056;
        double r117065 = 3.0;
        double r117066 = pow(r117064, r117065);
        double r117067 = r117063 * r117066;
        double r117068 = 0.5;
        double r117069 = 2.0;
        double r117070 = pow(r117056, r117069);
        double r117071 = cbrt(r117055);
        double r117072 = r117071 * r117071;
        double r117073 = pow(r117072, r117069);
        double r117074 = r117070 * r117073;
        double r117075 = pow(r117071, r117069);
        double r117076 = r117074 * r117075;
        double r117077 = r117068 * r117076;
        double r117078 = r117056 * r117055;
        double r117079 = r117077 + r117078;
        double r117080 = r117067 + r117079;
        double r117081 = r117048 * r117055;
        double r117082 = exp(r117081);
        double r117083 = r117082 - r117060;
        double r117084 = cbrt(r117083);
        double r117085 = r117084 * r117084;
        double r117086 = r117085 * r117084;
        double r117087 = r117080 * r117086;
        double r117088 = r117062 / r117087;
        double r117089 = exp(r117078);
        double r117090 = r117089 - r117060;
        double r117091 = pow(r117055, r117065);
        double r117092 = pow(r117048, r117065);
        double r117093 = r117091 * r117092;
        double r117094 = r117063 * r117093;
        double r117095 = pow(r117055, r117069);
        double r117096 = pow(r117048, r117069);
        double r117097 = r117095 * r117096;
        double r117098 = r117068 * r117097;
        double r117099 = r117055 * r117048;
        double r117100 = r117098 + r117099;
        double r117101 = r117094 + r117100;
        double r117102 = r117090 * r117101;
        double r117103 = r117062 / r117102;
        double r117104 = r117054 ? r117088 : r117103;
        return r117104;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.3
Target15.5
Herbie53.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -2.140357494799946e+116 or 7.604732750432952e-35 < b

    1. Initial program 55.5

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

    2. Taylor expanded around 0 49.2

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

    4. Applied pow-prod-down47.8

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

    5. Simplified47.8

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

    7. Applied add-cube-cbrt47.8

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

    8. Applied unpow-prod-down47.8

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

    9. Applied associate-*r*47.5

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

    11. Applied add-cube-cbrt47.6

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

    if -2.140357494799946e+116 < b < 7.604732750432952e-35

    1. Initial program 63.2

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

    2. Taylor expanded around 0 56.8

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

  4. Final simplification53.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1403574947999459 \cdot 10^{116} \lor \neg \left(b \le 7.604732750432952 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot \left(\left({a}^{2} \cdot {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{e^{b \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))