Average Error: 58.9 → 3.1
Time: 31.2s
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 r1877237 = eps;
        double r1877238 = a;
        double r1877239 = b;
        double r1877240 = r1877238 + r1877239;
        double r1877241 = r1877240 * r1877237;
        double r1877242 = exp(r1877241);
        double r1877243 = 1.0;
        double r1877244 = r1877242 - r1877243;
        double r1877245 = r1877237 * r1877244;
        double r1877246 = r1877238 * r1877237;
        double r1877247 = exp(r1877246);
        double r1877248 = r1877247 - r1877243;
        double r1877249 = r1877239 * r1877237;
        double r1877250 = exp(r1877249);
        double r1877251 = r1877250 - r1877243;
        double r1877252 = r1877248 * r1877251;
        double r1877253 = r1877245 / r1877252;
        return r1877253;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r1877254 = 1.0;
        double r1877255 = a;
        double r1877256 = r1877254 / r1877255;
        double r1877257 = b;
        double r1877258 = r1877254 / r1877257;
        double r1877259 = r1877256 + r1877258;
        return r1877259;
}

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
Target14.1
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. Simplified39.4

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

  3. Taylor expanded around 0 3.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2019151 +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))))