Average Error: 58.8 → 3.2
Time: 4.4m
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 r3850986 = eps;
        double r3850987 = a;
        double r3850988 = b;
        double r3850989 = r3850987 + r3850988;
        double r3850990 = r3850989 * r3850986;
        double r3850991 = exp(r3850990);
        double r3850992 = 1.0;
        double r3850993 = r3850991 - r3850992;
        double r3850994 = r3850986 * r3850993;
        double r3850995 = r3850987 * r3850986;
        double r3850996 = exp(r3850995);
        double r3850997 = r3850996 - r3850992;
        double r3850998 = r3850988 * r3850986;
        double r3850999 = exp(r3850998);
        double r3851000 = r3850999 - r3850992;
        double r3851001 = r3850997 * r3851000;
        double r3851002 = r3850994 / r3851001;
        return r3851002;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3851003 = 1.0;
        double r3851004 = a;
        double r3851005 = r3851003 / r3851004;
        double r3851006 = b;
        double r3851007 = r3851003 / r3851006;
        double r3851008 = r3851005 + r3851007;
        return r3851008;
}

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

Original58.8
Target13.9
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.8

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

  2. Taylor expanded around 0 56.5

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

  3. Simplified55.4

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

  4. Taylor expanded around 0 3.2

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

  5. Final simplification3.2

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

Reproduce

herbie shell --seed 2019158 
(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))))