Average Error: 58.9 → 3.1
Time: 52.0s
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 r7415083 = eps;
        double r7415084 = a;
        double r7415085 = b;
        double r7415086 = r7415084 + r7415085;
        double r7415087 = r7415086 * r7415083;
        double r7415088 = exp(r7415087);
        double r7415089 = 1.0;
        double r7415090 = r7415088 - r7415089;
        double r7415091 = r7415083 * r7415090;
        double r7415092 = r7415084 * r7415083;
        double r7415093 = exp(r7415092);
        double r7415094 = r7415093 - r7415089;
        double r7415095 = r7415085 * r7415083;
        double r7415096 = exp(r7415095);
        double r7415097 = r7415096 - r7415089;
        double r7415098 = r7415094 * r7415097;
        double r7415099 = r7415091 / r7415098;
        return r7415099;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r7415100 = 1.0;
        double r7415101 = a;
        double r7415102 = r7415100 / r7415101;
        double r7415103 = b;
        double r7415104 = r7415100 / r7415103;
        double r7415105 = r7415102 + r7415104;
        return r7415105;
}

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
Target13.7
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 56.1

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

  3. Simplified55.1

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

  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 2019107 
(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))))