Average Error: 58.4 → 3.6
Time: 41.9s
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 r3991820 = eps;
        double r3991821 = a;
        double r3991822 = b;
        double r3991823 = r3991821 + r3991822;
        double r3991824 = r3991823 * r3991820;
        double r3991825 = exp(r3991824);
        double r3991826 = 1.0;
        double r3991827 = r3991825 - r3991826;
        double r3991828 = r3991820 * r3991827;
        double r3991829 = r3991821 * r3991820;
        double r3991830 = exp(r3991829);
        double r3991831 = r3991830 - r3991826;
        double r3991832 = r3991822 * r3991820;
        double r3991833 = exp(r3991832);
        double r3991834 = r3991833 - r3991826;
        double r3991835 = r3991831 * r3991834;
        double r3991836 = r3991828 / r3991835;
        return r3991836;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3991837 = 1.0;
        double r3991838 = a;
        double r3991839 = r3991837 / r3991838;
        double r3991840 = b;
        double r3991841 = r3991837 / r3991840;
        double r3991842 = r3991839 + r3991841;
        return r3991842;
}

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

Original58.4
Target14.5
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 56.3

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

  3. Simplified55.1

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

  4. Taylor expanded around 0 3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2019134 
(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))))