Average Error: 60.2 → 3.5
Time: 10.7s
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 r76472 = eps;
        double r76473 = a;
        double r76474 = b;
        double r76475 = r76473 + r76474;
        double r76476 = r76475 * r76472;
        double r76477 = exp(r76476);
        double r76478 = 1.0;
        double r76479 = r76477 - r76478;
        double r76480 = r76472 * r76479;
        double r76481 = r76473 * r76472;
        double r76482 = exp(r76481);
        double r76483 = r76482 - r76478;
        double r76484 = r76474 * r76472;
        double r76485 = exp(r76484);
        double r76486 = r76485 - r76478;
        double r76487 = r76483 * r76486;
        double r76488 = r76480 / r76487;
        return r76488;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r76489 = 1.0;
        double r76490 = b;
        double r76491 = r76489 / r76490;
        double r76492 = a;
        double r76493 = r76489 / r76492;
        double r76494 = r76491 + r76493;
        return r76494;
}

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

Original60.2
Target14.8
Herbie3.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 3.5

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

  3. Final simplification3.5

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

Reproduce

herbie shell --seed 2020034 
(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))))