Average Error: 60.4 → 3.3
Time: 30.5s
Precision: 64
\[-1 \lt \varepsilon \land \varepsilon \lt 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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 r166236 = eps;
        double r166237 = a;
        double r166238 = b;
        double r166239 = r166237 + r166238;
        double r166240 = r166239 * r166236;
        double r166241 = exp(r166240);
        double r166242 = 1.0;
        double r166243 = r166241 - r166242;
        double r166244 = r166236 * r166243;
        double r166245 = r166237 * r166236;
        double r166246 = exp(r166245);
        double r166247 = r166246 - r166242;
        double r166248 = r166238 * r166236;
        double r166249 = exp(r166248);
        double r166250 = r166249 - r166242;
        double r166251 = r166247 * r166250;
        double r166252 = r166244 / r166251;
        return r166252;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r166253 = 1.0;
        double r166254 = b;
        double r166255 = r166253 / r166254;
        double r166256 = a;
        double r166257 = r166253 / r166256;
        double r166258 = r166255 + r166257;
        return r166258;
}

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
Target14.9
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

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

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

  3. Simplified58.1

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

  4. Taylor expanded around 0 3.3

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

  5. Final simplification3.3

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))