Average Error: 60.3 → 3.5
Time: 27.0s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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)}
↓
\frac{1}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r42240 = eps;
        double r42241 = a;
        double r42242 = b;
        double r42243 = r42241 + r42242;
        double r42244 = r42243 * r42240;
        double r42245 = exp(r42244);
        double r42246 = 1.0;
        double r42247 = r42245 - r42246;
        double r42248 = r42240 * r42247;
        double r42249 = r42241 * r42240;
        double r42250 = exp(r42249);
        double r42251 = r42250 - r42246;
        double r42252 = r42242 * r42240;
        double r42253 = exp(r42252);
        double r42254 = r42253 - r42246;
        double r42255 = r42251 * r42254;
        double r42256 = r42248 / r42255;
        return r42256;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r42257 = 1.0;
        double r42258 = a;
        double r42259 = r42257 / r42258;
        double r42260 = b;
        double r42261 = r42257 / r42260;
        double r42262 = r42259 + r42261;
        return r42262;
}

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
Target15.0
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Taylor expanded around 0 57.8

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

  3. Simplified57.8

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

  4. Taylor expanded around 0 3.5

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

  5. Final simplification3.5

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

Reproduce

herbie shell --seed 2019174 
(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))))