Average Error: 60.3 → 3.4
Time: 1.0m
Precision: 64
\[-1.0 \lt \varepsilon \land \varepsilon \lt 1.0\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \left(e^{b \cdot \varepsilon} - 1.0\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \left(e^{b \cdot \varepsilon} - 1.0\right)}
↓
\frac{1}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r6072275 = eps;
        double r6072276 = a;
        double r6072277 = b;
        double r6072278 = r6072276 + r6072277;
        double r6072279 = r6072278 * r6072275;
        double r6072280 = exp(r6072279);
        double r6072281 = 1.0;
        double r6072282 = r6072280 - r6072281;
        double r6072283 = r6072275 * r6072282;
        double r6072284 = r6072276 * r6072275;
        double r6072285 = exp(r6072284);
        double r6072286 = r6072285 - r6072281;
        double r6072287 = r6072277 * r6072275;
        double r6072288 = exp(r6072287);
        double r6072289 = r6072288 - r6072281;
        double r6072290 = r6072286 * r6072289;
        double r6072291 = r6072283 / r6072290;
        return r6072291;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r6072292 = 1.0;
        double r6072293 = a;
        double r6072294 = r6072292 / r6072293;
        double r6072295 = b;
        double r6072296 = r6072292 / r6072295;
        double r6072297 = r6072294 + r6072296;
        return r6072297;
}

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
Target14.6
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

    \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \left(e^{b \cdot \varepsilon} - 1.0\right)}\]

  2. Taylor expanded around 0 57.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}}\]

  3. Simplified57.6

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1.0\right)}{\left(e^{a \cdot \varepsilon} - 1.0\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(\varepsilon \cdot b + \left(\left(b \cdot \varepsilon\right) \cdot \left(b \cdot \varepsilon\right)\right) \cdot \frac{1}{2}\right)\right)}}\]

  4. Taylor expanded around 0 3.4

    \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]

  5. Final simplification3.4

    \[\leadsto \frac{1}{a} + \frac{1}{b}\]

Reproduce

herbie shell --seed 2019165 
(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))))