Average Error: 60.4 → 3.2
Time: 18.1s
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 r104472 = eps;
        double r104473 = a;
        double r104474 = b;
        double r104475 = r104473 + r104474;
        double r104476 = r104475 * r104472;
        double r104477 = exp(r104476);
        double r104478 = 1.0;
        double r104479 = r104477 - r104478;
        double r104480 = r104472 * r104479;
        double r104481 = r104473 * r104472;
        double r104482 = exp(r104481);
        double r104483 = r104482 - r104478;
        double r104484 = r104474 * r104472;
        double r104485 = exp(r104484);
        double r104486 = r104485 - r104478;
        double r104487 = r104483 * r104486;
        double r104488 = r104480 / r104487;
        return r104488;
}
↓
double f(double a, double b, double __attribute__((unused)) eps) {
        double r104489 = 1.0;
        double r104490 = b;
        double r104491 = r104489 / r104490;
        double r104492 = a;
        double r104493 = r104489 / r104492;
        double r104494 = r104491 + r104493;
        return r104494;
}

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

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 3.2

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

  3. Final simplification3.2

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))