expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 9.1s

Precision: 64

• could not determine a ground truth for program body

  1. a = 3.8117447302796514e+224
  2. b = -1.1357936159238701e-278
  3. eps = 1.4280211935663203e-186

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

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 r91759 = eps;
        double r91760 = a;
        double r91761 = b;
        double r91762 = r91760 + r91761;
        double r91763 = r91762 * r91759;
        double r91764 = exp(r91763);
        double r91765 = 1.0;
        double r91766 = r91764 - r91765;
        double r91767 = r91759 * r91766;
        double r91768 = r91760 * r91759;
        double r91769 = exp(r91768);
        double r91770 = r91769 - r91765;
        double r91771 = r91761 * r91759;
        double r91772 = exp(r91771);
        double r91773 = r91772 - r91765;
        double r91774 = r91770 * r91773;
        double r91775 = r91767 / r91774;
        return r91775;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r91776 = 1.0;
        double r91777 = b;
        double r91778 = r91776 / r91777;
        double r91779 = a;
        double r91780 = r91776 / r91779;
        double r91781 = r91778 + r91780;
        return r91781;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.2
Herbie	3.4

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

Derivation

1. Initial program 60.3

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

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 2020049 +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))))