Average Error: 58.7 → 3.3
Time: 37.0s
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 r4882978 = eps;
        double r4882979 = a;
        double r4882980 = b;
        double r4882981 = r4882979 + r4882980;
        double r4882982 = r4882981 * r4882978;
        double r4882983 = exp(r4882982);
        double r4882984 = 1.0;
        double r4882985 = r4882983 - r4882984;
        double r4882986 = r4882978 * r4882985;
        double r4882987 = r4882979 * r4882978;
        double r4882988 = exp(r4882987);
        double r4882989 = r4882988 - r4882984;
        double r4882990 = r4882980 * r4882978;
        double r4882991 = exp(r4882990);
        double r4882992 = r4882991 - r4882984;
        double r4882993 = r4882989 * r4882992;
        double r4882994 = r4882986 / r4882993;
        return r4882994;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4882995 = 1.0;
        double r4882996 = a;
        double r4882997 = r4882995 / r4882996;
        double r4882998 = b;
        double r4882999 = r4882995 / r4882998;
        double r4883000 = r4882997 + r4882999;
        return r4883000;
}

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

Original58.7
Target14.5
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.7

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

  2. Taylor expanded around 0 3.3

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

  3. Final simplification3.3

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

Reproduce

herbie shell --seed 2019162 
(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))))