Average Error: 58.7 → 3.3
Time: 28.4s
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 r1551740 = eps;
        double r1551741 = a;
        double r1551742 = b;
        double r1551743 = r1551741 + r1551742;
        double r1551744 = r1551743 * r1551740;
        double r1551745 = exp(r1551744);
        double r1551746 = 1.0;
        double r1551747 = r1551745 - r1551746;
        double r1551748 = r1551740 * r1551747;
        double r1551749 = r1551741 * r1551740;
        double r1551750 = exp(r1551749);
        double r1551751 = r1551750 - r1551746;
        double r1551752 = r1551742 * r1551740;
        double r1551753 = exp(r1551752);
        double r1551754 = r1551753 - r1551746;
        double r1551755 = r1551751 * r1551754;
        double r1551756 = r1551748 / r1551755;
        return r1551756;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r1551757 = 1.0;
        double r1551758 = a;
        double r1551759 = r1551757 / r1551758;
        double r1551760 = b;
        double r1551761 = r1551757 / r1551760;
        double r1551762 = r1551759 + r1551761;
        return r1551762;
}

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
Target14.3
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 2019156 
(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))))