Average Error: 58.8 → 3.2
Time: 40.6s
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 r2381072 = eps;
        double r2381073 = a;
        double r2381074 = b;
        double r2381075 = r2381073 + r2381074;
        double r2381076 = r2381075 * r2381072;
        double r2381077 = exp(r2381076);
        double r2381078 = 1.0;
        double r2381079 = r2381077 - r2381078;
        double r2381080 = r2381072 * r2381079;
        double r2381081 = r2381073 * r2381072;
        double r2381082 = exp(r2381081);
        double r2381083 = r2381082 - r2381078;
        double r2381084 = r2381074 * r2381072;
        double r2381085 = exp(r2381084);
        double r2381086 = r2381085 - r2381078;
        double r2381087 = r2381083 * r2381086;
        double r2381088 = r2381080 / r2381087;
        return r2381088;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r2381089 = 1.0;
        double r2381090 = a;
        double r2381091 = r2381089 / r2381090;
        double r2381092 = b;
        double r2381093 = r2381089 / r2381092;
        double r2381094 = r2381091 + r2381093;
        return r2381094;
}

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 56.3

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

  3. Simplified56.2

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

  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))))