Average Error: 60.4 → 3.3
Time: 29.5s
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 r99092 = eps;
        double r99093 = a;
        double r99094 = b;
        double r99095 = r99093 + r99094;
        double r99096 = r99095 * r99092;
        double r99097 = exp(r99096);
        double r99098 = 1.0;
        double r99099 = r99097 - r99098;
        double r99100 = r99092 * r99099;
        double r99101 = r99093 * r99092;
        double r99102 = exp(r99101);
        double r99103 = r99102 - r99098;
        double r99104 = r99094 * r99092;
        double r99105 = exp(r99104);
        double r99106 = r99105 - r99098;
        double r99107 = r99103 * r99106;
        double r99108 = r99100 / r99107;
        return r99108;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r99109 = 1.0;
        double r99110 = b;
        double r99111 = r99109 / r99110;
        double r99112 = a;
        double r99113 = r99109 / r99112;
        double r99114 = r99111 + r99113;
        return r99114;
}

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

Original60.4
Target15.0
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.4

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

  2. Taylor expanded around 0 58.0

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

  4. Applied pow-prod-down57.5

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

  5. Simplified57.5

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

  6. Taylor expanded around 0 3.3

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

  7. Final simplification3.3

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

Reproduce

herbie shell --seed 2019212 
(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))))