Average Error: 60.3 → 53.3
Time: 12.3s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1403574947999459 \cdot 10^{116} \lor \neg \left(b \le 7.604732750432952 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{6} \cdot {\left(\varepsilon \cdot a\right)}^{3} + \left(\frac{1}{2} \cdot \left(\left({a}^{2} \cdot {\left(\sqrt[3]{\varepsilon} \cdot \sqrt[3]{\varepsilon}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\varepsilon}\right)}^{2}\right) + a \cdot \varepsilon\right)\right) \cdot \left(\left(\sqrt[3]{e^{b \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{b \cdot \varepsilon} - 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}\\ \end{array}\]
\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 r99303 = eps;
        double r99304 = a;
        double r99305 = b;
        double r99306 = r99304 + r99305;
        double r99307 = r99306 * r99303;
        double r99308 = exp(r99307);
        double r99309 = 1.0;
        double r99310 = r99308 - r99309;
        double r99311 = r99303 * r99310;
        double r99312 = r99304 * r99303;
        double r99313 = exp(r99312);
        double r99314 = r99313 - r99309;
        double r99315 = r99305 * r99303;
        double r99316 = exp(r99315);
        double r99317 = r99316 - r99309;
        double r99318 = r99314 * r99317;
        double r99319 = r99311 / r99318;
        return r99319;
}
↓
double f(double a, double b, double eps) {
        double r99320 = b;
        double r99321 = -2.140357494799946e+116;
        bool r99322 = r99320 <= r99321;
        double r99323 = 7.604732750432952e-35;
        bool r99324 = r99320 <= r99323;
        double r99325 = !r99324;
        bool r99326 = r99322 || r99325;
        double r99327 = eps;
        double r99328 = a;
        double r99329 = r99328 + r99320;
        double r99330 = r99329 * r99327;
        double r99331 = exp(r99330);
        double r99332 = 1.0;
        double r99333 = r99331 - r99332;
        double r99334 = r99327 * r99333;
        double r99335 = 0.16666666666666666;
        double r99336 = r99327 * r99328;
        double r99337 = 3.0;
        double r99338 = pow(r99336, r99337);
        double r99339 = r99335 * r99338;
        double r99340 = 0.5;
        double r99341 = 2.0;
        double r99342 = pow(r99328, r99341);
        double r99343 = cbrt(r99327);
        double r99344 = r99343 * r99343;
        double r99345 = pow(r99344, r99341);
        double r99346 = r99342 * r99345;
        double r99347 = pow(r99343, r99341);
        double r99348 = r99346 * r99347;
        double r99349 = r99340 * r99348;
        double r99350 = r99328 * r99327;
        double r99351 = r99349 + r99350;
        double r99352 = r99339 + r99351;
        double r99353 = r99320 * r99327;
        double r99354 = exp(r99353);
        double r99355 = r99354 - r99332;
        double r99356 = cbrt(r99355);
        double r99357 = r99356 * r99356;
        double r99358 = r99357 * r99356;
        double r99359 = r99352 * r99358;
        double r99360 = r99334 / r99359;
        double r99361 = exp(r99350);
        double r99362 = r99361 - r99332;
        double r99363 = pow(r99327, r99337);
        double r99364 = pow(r99320, r99337);
        double r99365 = r99363 * r99364;
        double r99366 = r99335 * r99365;
        double r99367 = pow(r99327, r99341);
        double r99368 = pow(r99320, r99341);
        double r99369 = r99367 * r99368;
        double r99370 = r99340 * r99369;
        double r99371 = r99327 * r99320;
        double r99372 = r99370 + r99371;
        double r99373 = r99366 + r99372;
        double r99374 = r99362 * r99373;
        double r99375 = r99334 / r99374;
        double r99376 = r99326 ? r99360 : r99375;
        return r99376;
}

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))))