Average Error: 58.6 → 3.3
Time: 53.5s
Precision: 64
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \frac{1}{b}\]
double f(double a, double b, double eps) {
        double r9765294 = eps;
        double r9765295 = a;
        double r9765296 = b;
        double r9765297 = r9765295 + r9765296;
        double r9765298 = r9765297 * r9765294;
        double r9765299 = exp(r9765298);
        double r9765300 = 1.0;
        double r9765301 = r9765299 - r9765300;
        double r9765302 = r9765294 * r9765301;
        double r9765303 = r9765295 * r9765294;
        double r9765304 = exp(r9765303);
        double r9765305 = r9765304 - r9765300;
        double r9765306 = r9765296 * r9765294;
        double r9765307 = exp(r9765306);
        double r9765308 = r9765307 - r9765300;
        double r9765309 = r9765305 * r9765308;
        double r9765310 = r9765302 / r9765309;
        return r9765310;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r9765311 = 1.0;
        double r9765312 = a;
        double r9765313 = r9765311 / r9765312;
        double r9765314 = b;
        double r9765315 = r9765311 / r9765314;
        double r9765316 = r9765313 + r9765315;
        return r9765316;
}


\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}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.1
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.6

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

  2. Simplified34.0

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

  3. Taylor expanded around 0 3.3

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

  4. Final simplification3.3

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(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))))