Average Error: 58.5 → 3.4
Time: 43.4s
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 r3483933 = eps;
        double r3483934 = a;
        double r3483935 = b;
        double r3483936 = r3483934 + r3483935;
        double r3483937 = r3483936 * r3483933;
        double r3483938 = exp(r3483937);
        double r3483939 = 1.0;
        double r3483940 = r3483938 - r3483939;
        double r3483941 = r3483933 * r3483940;
        double r3483942 = r3483934 * r3483933;
        double r3483943 = exp(r3483942);
        double r3483944 = r3483943 - r3483939;
        double r3483945 = r3483935 * r3483933;
        double r3483946 = exp(r3483945);
        double r3483947 = r3483946 - r3483939;
        double r3483948 = r3483944 * r3483947;
        double r3483949 = r3483941 / r3483948;
        return r3483949;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r3483950 = 1.0;
        double r3483951 = a;
        double r3483952 = r3483950 / r3483951;
        double r3483953 = b;
        double r3483954 = r3483950 / r3483953;
        double r3483955 = r3483952 + r3483954;
        return r3483955;
}

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

Original58.5
Target13.9
Herbie3.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 3.4

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

  3. Final simplification3.4

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

Reproduce

herbie shell --seed 2019138 
(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))))