Average Error: 60.3 → 3.4
Time: 10.9s
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 r88217 = eps;
        double r88218 = a;
        double r88219 = b;
        double r88220 = r88218 + r88219;
        double r88221 = r88220 * r88217;
        double r88222 = exp(r88221);
        double r88223 = 1.0;
        double r88224 = r88222 - r88223;
        double r88225 = r88217 * r88224;
        double r88226 = r88218 * r88217;
        double r88227 = exp(r88226);
        double r88228 = r88227 - r88223;
        double r88229 = r88219 * r88217;
        double r88230 = exp(r88229);
        double r88231 = r88230 - r88223;
        double r88232 = r88228 * r88231;
        double r88233 = r88225 / r88232;
        return r88233;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r88234 = 1.0;
        double r88235 = b;
        double r88236 = r88234 / r88235;
        double r88237 = a;
        double r88238 = r88234 / r88237;
        double r88239 = r88236 + r88238;
        return r88239;
}

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

Original60.3
Target14.7
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.3

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

  2. Taylor expanded around 0 57.8

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

  4. Applied pow-prod-down57.3

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

  5. Simplified57.3

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

  6. Taylor expanded around 0 3.4

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

  7. Final simplification3.4

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

Reproduce

herbie shell --seed 2020081 
(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))))