expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 8.8s

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 r98620 = eps;
        double r98621 = a;
        double r98622 = b;
        double r98623 = r98621 + r98622;
        double r98624 = r98623 * r98620;
        double r98625 = exp(r98624);
        double r98626 = 1.0;
        double r98627 = r98625 - r98626;
        double r98628 = r98620 * r98627;
        double r98629 = r98621 * r98620;
        double r98630 = exp(r98629);
        double r98631 = r98630 - r98626;
        double r98632 = r98622 * r98620;
        double r98633 = exp(r98632);
        double r98634 = r98633 - r98626;
        double r98635 = r98631 * r98634;
        double r98636 = r98628 / r98635;
        return r98636;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r98637 = 1.0;
        double r98638 = b;
        double r98639 = r98637 / r98638;
        double r98640 = a;
        double r98641 = r98637 / r98640;
        double r98642 = r98639 + r98641;
        return r98642;
}

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 
(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))))