expq3 (problem 3.4.2)

Average Error: 60.4 → 3.3

Time: 10.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = -2.0882778719170116e-105
  2. b = -1.1140740905072972e+272
  3. eps = -8.795263419266061e-104

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

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 r92452 = eps;
        double r92453 = a;
        double r92454 = b;
        double r92455 = r92453 + r92454;
        double r92456 = r92455 * r92452;
        double r92457 = exp(r92456);
        double r92458 = 1.0;
        double r92459 = r92457 - r92458;
        double r92460 = r92452 * r92459;
        double r92461 = r92453 * r92452;
        double r92462 = exp(r92461);
        double r92463 = r92462 - r92458;
        double r92464 = r92454 * r92452;
        double r92465 = exp(r92464);
        double r92466 = r92465 - r92458;
        double r92467 = r92463 * r92466;
        double r92468 = r92460 / r92467;
        return r92468;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r92469 = 1.0;
        double r92470 = b;
        double r92471 = r92469 / r92470;
        double r92472 = a;
        double r92473 = r92469 / r92472;
        double r92474 = r92471 + r92473;
        return r92474;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	15.2
Herbie	3.3

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

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 3.3

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

3. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019362 +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))))