Average Error: 60.3 → 53.3
Time: 13.6s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1403574947999459 \cdot 10^{116} \lor \neg \left(b \le 7.604732750432952 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {\left(\varepsilon \cdot a\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \left({a}^{2} \cdot {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{e^{b \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \end{array}\]
\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)}{\mathsf{fma}\left(\frac{1}{6}, {\left(\varepsilon \cdot a\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, \left({a}^{2} \cdot {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{2}, 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 \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\

\end{array}
double f(double a, double b, double eps) {
        double r137160 = eps;
        double r137161 = a;
        double r137162 = b;
        double r137163 = r137161 + r137162;
        double r137164 = r137163 * r137160;
        double r137165 = exp(r137164);
        double r137166 = 1.0;
        double r137167 = r137165 - r137166;
        double r137168 = r137160 * r137167;
        double r137169 = r137161 * r137160;
        double r137170 = exp(r137169);
        double r137171 = r137170 - r137166;
        double r137172 = r137162 * r137160;
        double r137173 = exp(r137172);
        double r137174 = r137173 - r137166;
        double r137175 = r137171 * r137174;
        double r137176 = r137168 / r137175;
        return r137176;
}
↓
double f(double a, double b, double eps) {
        double r137177 = b;
        double r137178 = -2.140357494799946e+116;
        bool r137179 = r137177 <= r137178;
        double r137180 = 7.604732750432952e-35;
        bool r137181 = r137177 <= r137180;
        double r137182 = !r137181;
        bool r137183 = r137179 || r137182;
        double r137184 = eps;
        double r137185 = a;
        double r137186 = r137185 + r137177;
        double r137187 = r137186 * r137184;
        double r137188 = exp(r137187);
        double r137189 = 1.0;
        double r137190 = r137188 - r137189;
        double r137191 = r137184 * r137190;
        double r137192 = 0.16666666666666666;
        double r137193 = r137184 * r137185;
        double r137194 = 3.0;
        double r137195 = pow(r137193, r137194);
        double r137196 = 0.5;
        double r137197 = 2.0;
        double r137198 = pow(r137185, r137197);
        double r137199 = cbrt(r137184);
        double r137200 = r137199 * r137199;
        double r137201 = pow(r137200, r137197);
        double r137202 = r137198 * r137201;
        double r137203 = pow(r137199, r137197);
        double r137204 = r137202 * r137203;
        double r137205 = r137185 * r137184;
        double r137206 = fma(r137196, r137204, r137205);
        double r137207 = fma(r137192, r137195, r137206);
        double r137208 = r137177 * r137184;
        double r137209 = exp(r137208);
        double r137210 = r137209 - r137189;
        double r137211 = cbrt(r137210);
        double r137212 = r137211 * r137211;
        double r137213 = r137212 * r137211;
        double r137214 = r137207 * r137213;
        double r137215 = r137191 / r137214;
        double r137216 = exp(r137205);
        double r137217 = r137216 - r137189;
        double r137218 = pow(r137184, r137194);
        double r137219 = pow(r137177, r137194);
        double r137220 = r137218 * r137219;
        double r137221 = pow(r137184, r137197);
        double r137222 = pow(r137177, r137197);
        double r137223 = r137221 * r137222;
        double r137224 = r137184 * r137177;
        double r137225 = fma(r137196, r137223, r137224);
        double r137226 = fma(r137192, r137220, r137225);
        double r137227 = r137217 * r137226;
        double r137228 = r137191 / r137227;
        double r137229 = r137183 ? r137215 : r137228;
        return r137229;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

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. Simplified49.2

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

    5. Applied pow-prod-down47.8

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

    6. Simplified47.8

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

    8. Applied add-cube-cbrt47.8

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

    9. Applied unpow-prod-down47.8

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

    10. Applied associate-*r*47.5

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

    12. Applied add-cube-cbrt47.6

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

    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. Simplified56.8

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

  4. Final simplification53.3

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

Reproduce

herbie shell --seed 2020036 +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))))