expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 30.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.0312407122753113e+113
  2. b = 1.9702806022714736e+226
  3. eps = 1.5467071799125403e-134

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

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 r4403290 = eps;
        double r4403291 = a;
        double r4403292 = b;
        double r4403293 = r4403291 + r4403292;
        double r4403294 = r4403293 * r4403290;
        double r4403295 = exp(r4403294);
        double r4403296 = 1.0;
        double r4403297 = r4403295 - r4403296;
        double r4403298 = r4403290 * r4403297;
        double r4403299 = r4403291 * r4403290;
        double r4403300 = exp(r4403299);
        double r4403301 = r4403300 - r4403296;
        double r4403302 = r4403292 * r4403290;
        double r4403303 = exp(r4403302);
        double r4403304 = r4403303 - r4403296;
        double r4403305 = r4403301 * r4403304;
        double r4403306 = r4403298 / r4403305;
        return r4403306;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r4403307 = 1.0;
        double r4403308 = a;
        double r4403309 = r4403307 / r4403308;
        double r4403310 = b;
        double r4403311 = r4403307 / r4403310;
        double r4403312 = r4403309 + r4403311;
        return r4403312;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.9
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 2019179 +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))))