Average Error: 58.5 → 3.5
Time: 44.9s
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 r2527831 = eps;
        double r2527832 = a;
        double r2527833 = b;
        double r2527834 = r2527832 + r2527833;
        double r2527835 = r2527834 * r2527831;
        double r2527836 = exp(r2527835);
        double r2527837 = 1.0;
        double r2527838 = r2527836 - r2527837;
        double r2527839 = r2527831 * r2527838;
        double r2527840 = r2527832 * r2527831;
        double r2527841 = exp(r2527840);
        double r2527842 = r2527841 - r2527837;
        double r2527843 = r2527833 * r2527831;
        double r2527844 = exp(r2527843);
        double r2527845 = r2527844 - r2527837;
        double r2527846 = r2527842 * r2527845;
        double r2527847 = r2527839 / r2527846;
        return r2527847;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r2527848 = 1.0;
        double r2527849 = a;
        double r2527850 = r2527848 / r2527849;
        double r2527851 = b;
        double r2527852 = r2527848 / r2527851;
        double r2527853 = r2527850 + r2527852;
        return r2527853;
}

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

Original58.5
Target14.4
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 56.6

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

  3. Simplified55.3

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

  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 2019146 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))