Average Error: 60.6 → 3.2
Time: 12.1s
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 r96650 = eps;
        double r96651 = a;
        double r96652 = b;
        double r96653 = r96651 + r96652;
        double r96654 = r96653 * r96650;
        double r96655 = exp(r96654);
        double r96656 = 1.0;
        double r96657 = r96655 - r96656;
        double r96658 = r96650 * r96657;
        double r96659 = r96651 * r96650;
        double r96660 = exp(r96659);
        double r96661 = r96660 - r96656;
        double r96662 = r96652 * r96650;
        double r96663 = exp(r96662);
        double r96664 = r96663 - r96656;
        double r96665 = r96661 * r96664;
        double r96666 = r96658 / r96665;
        return r96666;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r96667 = 1.0;
        double r96668 = b;
        double r96669 = r96667 / r96668;
        double r96670 = a;
        double r96671 = r96667 / r96670;
        double r96672 = r96669 + r96671;
        return r96672;
}

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

Original60.6
Target14.7
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.6

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

  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 2020100 +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))))