Average Error: 58.3 → 3.6
Time: 35.6s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
\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 r2215447 = eps;
        double r2215448 = a;
        double r2215449 = b;
        double r2215450 = r2215448 + r2215449;
        double r2215451 = r2215450 * r2215447;
        double r2215452 = exp(r2215451);
        double r2215453 = 1.0;
        double r2215454 = r2215452 - r2215453;
        double r2215455 = r2215447 * r2215454;
        double r2215456 = r2215448 * r2215447;
        double r2215457 = exp(r2215456);
        double r2215458 = r2215457 - r2215453;
        double r2215459 = r2215449 * r2215447;
        double r2215460 = exp(r2215459);
        double r2215461 = r2215460 - r2215453;
        double r2215462 = r2215458 * r2215461;
        double r2215463 = r2215455 / r2215462;
        return r2215463;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r2215464 = 1.0;
        double r2215465 = a;
        double r2215466 = r2215464 / r2215465;
        double r2215467 = b;
        double r2215468 = r2215464 / r2215467;
        double r2215469 = r2215466 + r2215468;
        return r2215469;
}

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

Original58.3
Target14.5
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.3

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

  2. Simplified38.2

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

  3. Taylor expanded around 0 3.6

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

  4. Final simplification3.6

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

Reproduce

herbie shell --seed 2019133 +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))))