Average Error: 58.6 → 3.4
Time: 39.8s
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 r3673058 = eps;
        double r3673059 = a;
        double r3673060 = b;
        double r3673061 = r3673059 + r3673060;
        double r3673062 = r3673061 * r3673058;
        double r3673063 = exp(r3673062);
        double r3673064 = 1.0;
        double r3673065 = r3673063 - r3673064;
        double r3673066 = r3673058 * r3673065;
        double r3673067 = r3673059 * r3673058;
        double r3673068 = exp(r3673067);
        double r3673069 = r3673068 - r3673064;
        double r3673070 = r3673060 * r3673058;
        double r3673071 = exp(r3673070);
        double r3673072 = r3673071 - r3673064;
        double r3673073 = r3673069 * r3673072;
        double r3673074 = r3673066 / r3673073;
        return r3673074;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3673075 = 1.0;
        double r3673076 = a;
        double r3673077 = r3673075 / r3673076;
        double r3673078 = b;
        double r3673079 = r3673075 / r3673078;
        double r3673080 = r3673077 + r3673079;
        return r3673080;
}

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

Original58.6
Target14.5
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.6

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

  2. Simplified37.7

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

  3. Taylor expanded around 0 3.4

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

  4. Final simplification3.4

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

Reproduce

herbie shell --seed 2019130 +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))))