Average Error: 60.0 → 3.7
Time: 9.0s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r80854 = eps;
        double r80855 = a;
        double r80856 = b;
        double r80857 = r80855 + r80856;
        double r80858 = r80857 * r80854;
        double r80859 = exp(r80858);
        double r80860 = 1.0;
        double r80861 = r80859 - r80860;
        double r80862 = r80854 * r80861;
        double r80863 = r80855 * r80854;
        double r80864 = exp(r80863);
        double r80865 = r80864 - r80860;
        double r80866 = r80856 * r80854;
        double r80867 = exp(r80866);
        double r80868 = r80867 - r80860;
        double r80869 = r80865 * r80868;
        double r80870 = r80862 / r80869;
        return r80870;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r80871 = 1.0;
        double r80872 = b;
        double r80873 = r80871 / r80872;
        double r80874 = a;
        double r80875 = r80871 / r80874;
        double r80876 = r80873 + r80875;
        return r80876;
}

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

Original60.0
Target15.4
Herbie3.7
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.0

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

  2. Taylor expanded around 0 3.7

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

  3. Final simplification3.7

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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))))