Average Error: 58.4 → 3.5
Time: 1.4m
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 r12334529 = eps;
        double r12334530 = a;
        double r12334531 = b;
        double r12334532 = r12334530 + r12334531;
        double r12334533 = r12334532 * r12334529;
        double r12334534 = exp(r12334533);
        double r12334535 = 1.0;
        double r12334536 = r12334534 - r12334535;
        double r12334537 = r12334529 * r12334536;
        double r12334538 = r12334530 * r12334529;
        double r12334539 = exp(r12334538);
        double r12334540 = r12334539 - r12334535;
        double r12334541 = r12334531 * r12334529;
        double r12334542 = exp(r12334541);
        double r12334543 = r12334542 - r12334535;
        double r12334544 = r12334540 * r12334543;
        double r12334545 = r12334537 / r12334544;
        return r12334545;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r12334546 = 1.0;
        double r12334547 = a;
        double r12334548 = r12334546 / r12334547;
        double r12334549 = b;
        double r12334550 = r12334546 / r12334549;
        double r12334551 = r12334548 + r12334550;
        return r12334551;
}

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

Original58.4
Target14.2
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 58.4

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

  2. Simplified34.4

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}\]

  3. Taylor expanded around 0 3.5

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

  4. Final simplification3.5

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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))))