Average Error: 60.1 → 3.6
Time: 31.1s
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 r110063 = eps;
        double r110064 = a;
        double r110065 = b;
        double r110066 = r110064 + r110065;
        double r110067 = r110066 * r110063;
        double r110068 = exp(r110067);
        double r110069 = 1.0;
        double r110070 = r110068 - r110069;
        double r110071 = r110063 * r110070;
        double r110072 = r110064 * r110063;
        double r110073 = exp(r110072);
        double r110074 = r110073 - r110069;
        double r110075 = r110065 * r110063;
        double r110076 = exp(r110075);
        double r110077 = r110076 - r110069;
        double r110078 = r110074 * r110077;
        double r110079 = r110071 / r110078;
        return r110079;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r110080 = 1.0;
        double r110081 = a;
        double r110082 = r110080 / r110081;
        double r110083 = b;
        double r110084 = r110080 / r110083;
        double r110085 = r110082 + r110084;
        return r110085;
}

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

Original60.1
Target15.3
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.1

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

  2. Taylor expanded around 0 57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  3. Simplified57.9

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left({a}^{3} \cdot {\varepsilon}^{3}, \frac{1}{6}, \mathsf{fma}\left(\varepsilon, a, \left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  4. Taylor expanded around 0 3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))