expq3 (problem 3.4.2)

Average Error: 60.2 → 3.4

Time: 10.3s

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 r87612 = eps;
        double r87613 = a;
        double r87614 = b;
        double r87615 = r87613 + r87614;
        double r87616 = r87615 * r87612;
        double r87617 = exp(r87616);
        double r87618 = 1.0;
        double r87619 = r87617 - r87618;
        double r87620 = r87612 * r87619;
        double r87621 = r87613 * r87612;
        double r87622 = exp(r87621);
        double r87623 = r87622 - r87618;
        double r87624 = r87614 * r87612;
        double r87625 = exp(r87624);
        double r87626 = r87625 - r87618;
        double r87627 = r87623 * r87626;
        double r87628 = r87620 / r87627;
        return r87628;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r87629 = 1.0;
        double r87630 = b;
        double r87631 = r87629 / r87630;
        double r87632 = a;
        double r87633 = r87629 / r87632;
        double r87634 = r87631 + r87633;
        return r87634;
}

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 
(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))))