Average Error: 60.6 → 3.1
Time: 11.8s
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 r100954 = eps;
        double r100955 = a;
        double r100956 = b;
        double r100957 = r100955 + r100956;
        double r100958 = r100957 * r100954;
        double r100959 = exp(r100958);
        double r100960 = 1.0;
        double r100961 = r100959 - r100960;
        double r100962 = r100954 * r100961;
        double r100963 = r100955 * r100954;
        double r100964 = exp(r100963);
        double r100965 = r100964 - r100960;
        double r100966 = r100956 * r100954;
        double r100967 = exp(r100966);
        double r100968 = r100967 - r100960;
        double r100969 = r100965 * r100968;
        double r100970 = r100962 / r100969;
        return r100970;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r100971 = 1.0;
        double r100972 = b;
        double r100973 = r100971 / r100972;
        double r100974 = a;
        double r100975 = r100971 / r100974;
        double r100976 = r100973 + r100975;
        return r100976;
}

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

Original60.6
Target14.8
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.6

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

  2. Taylor expanded around 0 58.0

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

  4. Applied pow-prod-down57.6

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

  5. Simplified57.6

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

  6. Taylor expanded around 0 3.1

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

  7. Final simplification3.1

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

Reproduce

herbie shell --seed 2019352 
(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))))