Average Error: 60.2 → 52.0
Time: 2.3m
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.296149559147507139887653903723053266557 \cdot 10^{50}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1}\right) \cdot \sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1}\right) \cdot \varepsilon}{\left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)\right) \cdot \left(\left(b \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot b + b \cdot \varepsilon\right)}\\ \mathbf{elif}\;a \le 340393273.876882851123809814453125:\\ \;\;\;\;\frac{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \varepsilon}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot a + \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + \frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \varepsilon}{\left(\left(b \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot b + b \cdot \varepsilon\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \end{array}\]
\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}\;a \le -3.296149559147507139887653903723053266557 \cdot 10^{50}:\\
\;\;\;\;\frac{\left(\left(\sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1}\right) \cdot \sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1}\right) \cdot \varepsilon}{\left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)\right) \cdot \left(\left(b \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot b + b \cdot \varepsilon\right)}\\

\mathbf{elif}\;a \le 340393273.876882851123809814453125:\\
\;\;\;\;\frac{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \varepsilon}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot a + \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + \frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \varepsilon}{\left(\left(b \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot b + b \cdot \varepsilon\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\

\end{array}
double f(double a, double b, double eps) {
        double r6168256 = eps;
        double r6168257 = a;
        double r6168258 = b;
        double r6168259 = r6168257 + r6168258;
        double r6168260 = r6168259 * r6168256;
        double r6168261 = exp(r6168260);
        double r6168262 = 1.0;
        double r6168263 = r6168261 - r6168262;
        double r6168264 = r6168256 * r6168263;
        double r6168265 = r6168257 * r6168256;
        double r6168266 = exp(r6168265);
        double r6168267 = r6168266 - r6168262;
        double r6168268 = r6168258 * r6168256;
        double r6168269 = exp(r6168268);
        double r6168270 = r6168269 - r6168262;
        double r6168271 = r6168267 * r6168270;
        double r6168272 = r6168264 / r6168271;
        return r6168272;
}
↓
double f(double a, double b, double eps) {
        double r6168273 = a;
        double r6168274 = -3.296149559147507e+50;
        bool r6168275 = r6168273 <= r6168274;
        double r6168276 = eps;
        double r6168277 = b;
        double r6168278 = r6168277 + r6168273;
        double r6168279 = r6168276 * r6168278;
        double r6168280 = exp(r6168279);
        double r6168281 = 1.0;
        double r6168282 = r6168280 - r6168281;
        double r6168283 = cbrt(r6168282);
        double r6168284 = r6168283 * r6168283;
        double r6168285 = r6168284 * r6168283;
        double r6168286 = r6168285 * r6168276;
        double r6168287 = r6168276 * r6168273;
        double r6168288 = exp(r6168287);
        double r6168289 = r6168288 - r6168281;
        double r6168290 = cbrt(r6168289);
        double r6168291 = r6168290 * r6168290;
        double r6168292 = r6168290 * r6168291;
        double r6168293 = 0.16666666666666666;
        double r6168294 = r6168276 * r6168276;
        double r6168295 = r6168276 * r6168294;
        double r6168296 = r6168293 * r6168295;
        double r6168297 = r6168277 * r6168296;
        double r6168298 = 0.5;
        double r6168299 = r6168294 * r6168298;
        double r6168300 = r6168297 + r6168299;
        double r6168301 = r6168277 * r6168300;
        double r6168302 = r6168301 * r6168277;
        double r6168303 = r6168277 * r6168276;
        double r6168304 = r6168302 + r6168303;
        double r6168305 = r6168292 * r6168304;
        double r6168306 = r6168286 / r6168305;
        double r6168307 = 340393273.87688285;
        bool r6168308 = r6168273 <= r6168307;
        double r6168309 = r6168282 * r6168276;
        double r6168310 = exp(r6168303);
        double r6168311 = r6168310 - r6168281;
        double r6168312 = r6168273 * r6168273;
        double r6168313 = r6168273 * r6168312;
        double r6168314 = r6168293 * r6168313;
        double r6168315 = r6168276 * r6168314;
        double r6168316 = r6168298 * r6168312;
        double r6168317 = r6168315 + r6168316;
        double r6168318 = r6168317 * r6168294;
        double r6168319 = r6168287 + r6168318;
        double r6168320 = r6168311 * r6168319;
        double r6168321 = r6168309 / r6168320;
        double r6168322 = r6168304 * r6168289;
        double r6168323 = r6168309 / r6168322;
        double r6168324 = r6168308 ? r6168321 : r6168323;
        double r6168325 = r6168275 ? r6168306 : r6168324;
        return r6168325;
}

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

Original60.2
Target15.6
Herbie52.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -3.296149559147507e+50

    1. Initial program 54.5

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

    2. Taylor expanded around 0 47.3

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

    3. Simplified43.9

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

    5. Applied associate-*l*43.0

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

    7. Applied add-cube-cbrt43.0

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

    9. Applied add-cube-cbrt43.0

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

    if -3.296149559147507e+50 < a < 340393273.87688285

    1. Initial program 63.8

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

    2. Taylor expanded around 0 56.7

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

    3. Simplified56.7

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

    if 340393273.87688285 < a

    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 50.3

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

    3. Simplified47.9

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

    5. Applied associate-*l*46.9

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

  4. Final simplification52.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.296149559147507139887653903723053266557 \cdot 10^{50}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1}\right) \cdot \sqrt[3]{e^{\varepsilon \cdot \left(b + a\right)} - 1}\right) \cdot \varepsilon}{\left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \left(\sqrt[3]{e^{\varepsilon \cdot a} - 1} \cdot \sqrt[3]{e^{\varepsilon \cdot a} - 1}\right)\right) \cdot \left(\left(b \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot b + b \cdot \varepsilon\right)}\\ \mathbf{elif}\;a \le 340393273.876882851123809814453125:\\ \;\;\;\;\frac{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \varepsilon}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot a + \left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) + \frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\varepsilon \cdot \left(b + a\right)} - 1\right) \cdot \varepsilon}{\left(\left(b \cdot \left(b \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot b + b \cdot \varepsilon\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))