Average Error: 60.4 → 3.2
Time: 31.3s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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}{b} + \frac{1}{a}
double f(double a, double b, double eps) {
        double r80206 = eps;
        double r80207 = a;
        double r80208 = b;
        double r80209 = r80207 + r80208;
        double r80210 = r80209 * r80206;
        double r80211 = exp(r80210);
        double r80212 = 1.0;
        double r80213 = r80211 - r80212;
        double r80214 = r80206 * r80213;
        double r80215 = r80207 * r80206;
        double r80216 = exp(r80215);
        double r80217 = r80216 - r80212;
        double r80218 = r80208 * r80206;
        double r80219 = exp(r80218);
        double r80220 = r80219 - r80212;
        double r80221 = r80217 * r80220;
        double r80222 = r80214 / r80221;
        return r80222;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r80223 = 1.0;
        double r80224 = b;
        double r80225 = r80223 / r80224;
        double r80226 = a;
        double r80227 = r80223 / r80226;
        double r80228 = r80225 + r80227;
        return r80228;
}

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

Original60.4
Target15.1
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.4

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

  2. Taylor expanded around 0 58.2

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

  3. Simplified58.2

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

  4. Taylor expanded around 0 3.2

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

  5. Simplified3.2

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

  6. Final simplification3.2

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

Reproduce

herbie shell --seed 2019194 +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))))