Average Error: 60.5 → 3.2
Time: 32.1s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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}{b} + \frac{1}{a}
double f(double a, double b, double eps) {
        double r85833 = eps;
        double r85834 = a;
        double r85835 = b;
        double r85836 = r85834 + r85835;
        double r85837 = r85836 * r85833;
        double r85838 = exp(r85837);
        double r85839 = 1.0;
        double r85840 = r85838 - r85839;
        double r85841 = r85833 * r85840;
        double r85842 = r85834 * r85833;
        double r85843 = exp(r85842);
        double r85844 = r85843 - r85839;
        double r85845 = r85835 * r85833;
        double r85846 = exp(r85845);
        double r85847 = r85846 - r85839;
        double r85848 = r85844 * r85847;
        double r85849 = r85841 / r85848;
        return r85849;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r85850 = 1.0;
        double r85851 = b;
        double r85852 = r85850 / r85851;
        double r85853 = a;
        double r85854 = r85850 / r85853;
        double r85855 = r85852 + r85854;
        return r85855;
}

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

Original60.5
Target15.1
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.5

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

  2. Taylor expanded around 0 3.2

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

  3. Final simplification3.2

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

Reproduce

herbie shell --seed 2019322 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))