Average Error: 58.4 → 3.5
Time: 30.5s
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 r1895347 = eps;
        double r1895348 = a;
        double r1895349 = b;
        double r1895350 = r1895348 + r1895349;
        double r1895351 = r1895350 * r1895347;
        double r1895352 = exp(r1895351);
        double r1895353 = 1.0;
        double r1895354 = r1895352 - r1895353;
        double r1895355 = r1895347 * r1895354;
        double r1895356 = r1895348 * r1895347;
        double r1895357 = exp(r1895356);
        double r1895358 = r1895357 - r1895353;
        double r1895359 = r1895349 * r1895347;
        double r1895360 = exp(r1895359);
        double r1895361 = r1895360 - r1895353;
        double r1895362 = r1895358 * r1895361;
        double r1895363 = r1895355 / r1895362;
        return r1895363;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r1895364 = 1.0;
        double r1895365 = a;
        double r1895366 = r1895364 / r1895365;
        double r1895367 = b;
        double r1895368 = r1895364 / r1895367;
        double r1895369 = r1895366 + r1895368;
        return r1895369;
}

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

Original58.4
Target14.2
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.4

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

  2. Simplified27.7

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

  3. Taylor expanded around 0 3.5

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

  4. Final simplification3.5

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

Reproduce

herbie shell --seed 2019128 +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))))