Average Error: 60.2 → 3.5
Time: 9.1s
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 r90419 = eps;
        double r90420 = a;
        double r90421 = b;
        double r90422 = r90420 + r90421;
        double r90423 = r90422 * r90419;
        double r90424 = exp(r90423);
        double r90425 = 1.0;
        double r90426 = r90424 - r90425;
        double r90427 = r90419 * r90426;
        double r90428 = r90420 * r90419;
        double r90429 = exp(r90428);
        double r90430 = r90429 - r90425;
        double r90431 = r90421 * r90419;
        double r90432 = exp(r90431);
        double r90433 = r90432 - r90425;
        double r90434 = r90430 * r90433;
        double r90435 = r90427 / r90434;
        return r90435;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r90436 = 1.0;
        double r90437 = b;
        double r90438 = r90436 / r90437;
        double r90439 = a;
        double r90440 = r90436 / r90439;
        double r90441 = r90438 + r90440;
        return r90441;
}

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
Target14.8
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 2020034 +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))))