expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 34.8s

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 r30530 = eps;
        double r30531 = a;
        double r30532 = b;
        double r30533 = r30531 + r30532;
        double r30534 = r30533 * r30530;
        double r30535 = exp(r30534);
        double r30536 = 1.0;
        double r30537 = r30535 - r30536;
        double r30538 = r30530 * r30537;
        double r30539 = r30531 * r30530;
        double r30540 = exp(r30539);
        double r30541 = r30540 - r30536;
        double r30542 = r30532 * r30530;
        double r30543 = exp(r30542);
        double r30544 = r30543 - r30536;
        double r30545 = r30541 * r30544;
        double r30546 = r30538 / r30545;
        return r30546;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r30547 = 1.0;
        double r30548 = b;
        double r30549 = r30547 / r30548;
        double r30550 = a;
        double r30551 = r30547 / r30550;
        double r30552 = r30549 + r30551;
        return r30552;
}

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))))