expq3 (problem 3.4.2)

Average Error: 60.4 → 3.3

Time: 19.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. a = 7.130456307924188e+301
  2. b = -1.4864914298183363e-09
  3. eps = 3.907418602099771e-127

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

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 r109314 = eps;
        double r109315 = a;
        double r109316 = b;
        double r109317 = r109315 + r109316;
        double r109318 = r109317 * r109314;
        double r109319 = exp(r109318);
        double r109320 = 1.0;
        double r109321 = r109319 - r109320;
        double r109322 = r109314 * r109321;
        double r109323 = r109315 * r109314;
        double r109324 = exp(r109323);
        double r109325 = r109324 - r109320;
        double r109326 = r109316 * r109314;
        double r109327 = exp(r109326);
        double r109328 = r109327 - r109320;
        double r109329 = r109325 * r109328;
        double r109330 = r109322 / r109329;
        return r109330;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r109331 = 1.0;
        double r109332 = b;
        double r109333 = r109331 / r109332;
        double r109334 = a;
        double r109335 = r109331 / r109334;
        double r109336 = r109333 + r109335;
        return r109336;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.9
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 58.2

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]

3. Simplified 58.1

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(b \cdot \left(\varepsilon + \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right) \cdot b\right) + \frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)}}\]

4. Taylor expanded around 0 3.3

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

5. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019351 
(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))))