expq3 (problem 3.4.2)

Average Error: 60.2 → 3.4

Time: 10.6s

Precision: 64

• could not determine a ground truth for program body

  1. a = -2.147009436181225e-259
  2. b = 1.0551194375545526e+290
  3. eps = 3.8333244974596404e-56

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

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

↓

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

C

double f(double a, double b, double eps) {
        double r102440 = eps;
        double r102441 = a;
        double r102442 = b;
        double r102443 = r102441 + r102442;
        double r102444 = r102443 * r102440;
        double r102445 = exp(r102444);
        double r102446 = 1.0;
        double r102447 = r102445 - r102446;
        double r102448 = r102440 * r102447;
        double r102449 = r102441 * r102440;
        double r102450 = exp(r102449);
        double r102451 = r102450 - r102446;
        double r102452 = r102442 * r102440;
        double r102453 = exp(r102452);
        double r102454 = r102453 - r102446;
        double r102455 = r102451 * r102454;
        double r102456 = r102448 / r102455;
        return r102456;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r102457 = 1.0;
        double r102458 = b;
        double r102459 = r102457 / r102458;
        double r102460 = a;
        double r102461 = r102457 / r102460;
        double r102462 = r102459 + r102461;
        return r102462;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	14.7
Herbie	3.4

\[\frac{a + b}{a \cdot b}\]

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 3.4

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

3. Final simplification 3.4

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

Reproduce

herbie shell --seed 2020060 +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))))