expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 42.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.1856327996312082e-200
  2. b = -1.3185816627363143e+265
  3. eps = -4.093090337351439e-187

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

Math

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

↓

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

C

double f(double a, double b, double eps) {
        double r9474248 = eps;
        double r9474249 = a;
        double r9474250 = b;
        double r9474251 = r9474249 + r9474250;
        double r9474252 = r9474251 * r9474248;
        double r9474253 = exp(r9474252);
        double r9474254 = 1.0;
        double r9474255 = r9474253 - r9474254;
        double r9474256 = r9474248 * r9474255;
        double r9474257 = r9474249 * r9474248;
        double r9474258 = exp(r9474257);
        double r9474259 = r9474258 - r9474254;
        double r9474260 = r9474250 * r9474248;
        double r9474261 = exp(r9474260);
        double r9474262 = r9474261 - r9474254;
        double r9474263 = r9474259 * r9474262;
        double r9474264 = r9474256 / r9474263;
        return r9474264;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r9474265 = 1.0;
        double r9474266 = a;
        double r9474267 = r9474265 / r9474266;
        double r9474268 = b;
        double r9474269 = r9474265 / r9474268;
        double r9474270 = r9474267 + r9474269;
        return r9474270;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.5
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

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

3. Final simplification 3.4

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))