Average Error: 58.9 → 3.1
Time: 44.4s
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 r4456246 = eps;
        double r4456247 = a;
        double r4456248 = b;
        double r4456249 = r4456247 + r4456248;
        double r4456250 = r4456249 * r4456246;
        double r4456251 = exp(r4456250);
        double r4456252 = 1.0;
        double r4456253 = r4456251 - r4456252;
        double r4456254 = r4456246 * r4456253;
        double r4456255 = r4456247 * r4456246;
        double r4456256 = exp(r4456255);
        double r4456257 = r4456256 - r4456252;
        double r4456258 = r4456248 * r4456246;
        double r4456259 = exp(r4456258);
        double r4456260 = r4456259 - r4456252;
        double r4456261 = r4456257 * r4456260;
        double r4456262 = r4456254 / r4456261;
        return r4456262;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r4456263 = 1.0;
        double r4456264 = a;
        double r4456265 = r4456263 / r4456264;
        double r4456266 = b;
        double r4456267 = r4456263 / r4456266;
        double r4456268 = r4456265 + r4456267;
        return r4456268;
}

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

Original58.9
Target14.3
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.9

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

  2. Taylor expanded around 0 56.8

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

  3. Simplified55.7

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

  4. Taylor expanded around 0 3.1

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

  5. Final simplification3.1

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

Reproduce

herbie shell --seed 2019164 
(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))))