Average Error: 58.9 → 3.1
Time: 38.5s
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 r3391323 = eps;
        double r3391324 = a;
        double r3391325 = b;
        double r3391326 = r3391324 + r3391325;
        double r3391327 = r3391326 * r3391323;
        double r3391328 = exp(r3391327);
        double r3391329 = 1.0;
        double r3391330 = r3391328 - r3391329;
        double r3391331 = r3391323 * r3391330;
        double r3391332 = r3391324 * r3391323;
        double r3391333 = exp(r3391332);
        double r3391334 = r3391333 - r3391329;
        double r3391335 = r3391325 * r3391323;
        double r3391336 = exp(r3391335);
        double r3391337 = r3391336 - r3391329;
        double r3391338 = r3391334 * r3391337;
        double r3391339 = r3391331 / r3391338;
        return r3391339;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3391340 = 1.0;
        double r3391341 = a;
        double r3391342 = r3391340 / r3391341;
        double r3391343 = b;
        double r3391344 = r3391340 / r3391343;
        double r3391345 = r3391342 + r3391344;
        return r3391345;
}

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

Original58.9
Target14.3
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.9

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

  2. Simplified27.8

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

  3. Taylor expanded around 0 3.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2019135 +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))))