expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 33.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = -5.278031834180875e+72
  2. b = -5.9461326997729256e+41
  3. eps = -2.3299265771866867e-17

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

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 r91438 = eps;
        double r91439 = a;
        double r91440 = b;
        double r91441 = r91439 + r91440;
        double r91442 = r91441 * r91438;
        double r91443 = exp(r91442);
        double r91444 = 1.0;
        double r91445 = r91443 - r91444;
        double r91446 = r91438 * r91445;
        double r91447 = r91439 * r91438;
        double r91448 = exp(r91447);
        double r91449 = r91448 - r91444;
        double r91450 = r91440 * r91438;
        double r91451 = exp(r91450);
        double r91452 = r91451 - r91444;
        double r91453 = r91449 * r91452;
        double r91454 = r91446 / r91453;
        return r91454;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r91455 = 1.0;
        double r91456 = b;
        double r91457 = r91455 / r91456;
        double r91458 = a;
        double r91459 = r91455 / r91458;
        double r91460 = r91457 + r91459;
        return r91460;
}

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 2019323 +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))))