Average Error: 58.7 → 3.3
Time: 28.8s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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}{a} + \frac{1}{b}
double f(double a, double b, double eps) {
        double r2075485 = eps;
        double r2075486 = a;
        double r2075487 = b;
        double r2075488 = r2075486 + r2075487;
        double r2075489 = r2075488 * r2075485;
        double r2075490 = exp(r2075489);
        double r2075491 = 1.0;
        double r2075492 = r2075490 - r2075491;
        double r2075493 = r2075485 * r2075492;
        double r2075494 = r2075486 * r2075485;
        double r2075495 = exp(r2075494);
        double r2075496 = r2075495 - r2075491;
        double r2075497 = r2075487 * r2075485;
        double r2075498 = exp(r2075497);
        double r2075499 = r2075498 - r2075491;
        double r2075500 = r2075496 * r2075499;
        double r2075501 = r2075493 / r2075500;
        return r2075501;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r2075502 = 1.0;
        double r2075503 = a;
        double r2075504 = r2075502 / r2075503;
        double r2075505 = b;
        double r2075506 = r2075502 / r2075505;
        double r2075507 = r2075504 + r2075506;
        return r2075507;
}

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

Original58.7
Target14.3
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.7

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

  2. Simplified40.3

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

  3. Taylor expanded around 0 3.3

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))