Average Error: 60.2 → 3.5
Time: 10.3s
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 r94720 = eps;
        double r94721 = a;
        double r94722 = b;
        double r94723 = r94721 + r94722;
        double r94724 = r94723 * r94720;
        double r94725 = exp(r94724);
        double r94726 = 1.0;
        double r94727 = r94725 - r94726;
        double r94728 = r94720 * r94727;
        double r94729 = r94721 * r94720;
        double r94730 = exp(r94729);
        double r94731 = r94730 - r94726;
        double r94732 = r94722 * r94720;
        double r94733 = exp(r94732);
        double r94734 = r94733 - r94726;
        double r94735 = r94731 * r94734;
        double r94736 = r94728 / r94735;
        return r94736;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r94737 = 1.0;
        double r94738 = b;
        double r94739 = r94737 / r94738;
        double r94740 = a;
        double r94741 = r94737 / r94740;
        double r94742 = r94739 + r94741;
        return r94742;
}

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

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

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2020065 +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))))