Average Error: 60.2 → 3.5
Time: 9.3s
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 r90648 = eps;
        double r90649 = a;
        double r90650 = b;
        double r90651 = r90649 + r90650;
        double r90652 = r90651 * r90648;
        double r90653 = exp(r90652);
        double r90654 = 1.0;
        double r90655 = r90653 - r90654;
        double r90656 = r90648 * r90655;
        double r90657 = r90649 * r90648;
        double r90658 = exp(r90657);
        double r90659 = r90658 - r90654;
        double r90660 = r90650 * r90648;
        double r90661 = exp(r90660);
        double r90662 = r90661 - r90654;
        double r90663 = r90659 * r90662;
        double r90664 = r90656 / r90663;
        return r90664;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r90665 = 1.0;
        double r90666 = b;
        double r90667 = r90665 / r90666;
        double r90668 = a;
        double r90669 = r90665 / r90668;
        double r90670 = r90667 + r90669;
        return r90670;
}

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
Target15.2
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 2020064 
(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))))