Average Error: 58.8 → 3.2
Time: 37.7s
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 r3216221 = eps;
        double r3216222 = a;
        double r3216223 = b;
        double r3216224 = r3216222 + r3216223;
        double r3216225 = r3216224 * r3216221;
        double r3216226 = exp(r3216225);
        double r3216227 = 1.0;
        double r3216228 = r3216226 - r3216227;
        double r3216229 = r3216221 * r3216228;
        double r3216230 = r3216222 * r3216221;
        double r3216231 = exp(r3216230);
        double r3216232 = r3216231 - r3216227;
        double r3216233 = r3216223 * r3216221;
        double r3216234 = exp(r3216233);
        double r3216235 = r3216234 - r3216227;
        double r3216236 = r3216232 * r3216235;
        double r3216237 = r3216229 / r3216236;
        return r3216237;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3216238 = 1.0;
        double r3216239 = a;
        double r3216240 = r3216238 / r3216239;
        double r3216241 = b;
        double r3216242 = r3216238 / r3216241;
        double r3216243 = r3216240 + r3216242;
        return r3216243;
}

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

Original58.8
Target14.2
Herbie3.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.8

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

  2. Taylor expanded around 0 56.5

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

  3. Simplified55.5

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

  4. Taylor expanded around 0 3.2

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

  5. Final simplification3.2

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

Reproduce

herbie shell --seed 2019152 
(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))))