Average Error: 60.0 → 3.7
Time: 8.6s
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 r77511 = eps;
        double r77512 = a;
        double r77513 = b;
        double r77514 = r77512 + r77513;
        double r77515 = r77514 * r77511;
        double r77516 = exp(r77515);
        double r77517 = 1.0;
        double r77518 = r77516 - r77517;
        double r77519 = r77511 * r77518;
        double r77520 = r77512 * r77511;
        double r77521 = exp(r77520);
        double r77522 = r77521 - r77517;
        double r77523 = r77513 * r77511;
        double r77524 = exp(r77523);
        double r77525 = r77524 - r77517;
        double r77526 = r77522 * r77525;
        double r77527 = r77519 / r77526;
        return r77527;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r77528 = 1.0;
        double r77529 = b;
        double r77530 = r77528 / r77529;
        double r77531 = a;
        double r77532 = r77528 / r77531;
        double r77533 = r77530 + r77532;
        return r77533;
}

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 
(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))))