expq3 (problem 3.4.2)

Average Error: 60.5 → 3.2

Time: 21.7s

Precision: 64

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

• could not determine a ground truth for program body

  1. a = 1.3481840315279074e+121
  2. b = -7.480505059669827e-308
  3. eps = 3.5449819228153575e-79

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

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 r89226 = eps;
        double r89227 = a;
        double r89228 = b;
        double r89229 = r89227 + r89228;
        double r89230 = r89229 * r89226;
        double r89231 = exp(r89230);
        double r89232 = 1.0;
        double r89233 = r89231 - r89232;
        double r89234 = r89226 * r89233;
        double r89235 = r89227 * r89226;
        double r89236 = exp(r89235);
        double r89237 = r89236 - r89232;
        double r89238 = r89228 * r89226;
        double r89239 = exp(r89238);
        double r89240 = r89239 - r89232;
        double r89241 = r89237 * r89240;
        double r89242 = r89234 / r89241;
        return r89242;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r89243 = 1.0;
        double r89244 = b;
        double r89245 = r89243 / r89244;
        double r89246 = a;
        double r89247 = r89243 / r89246;
        double r89248 = r89245 + r89247;
        return r89248;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	15.0
Herbie	3.2

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

Derivation

1. Initial program 60.5

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

2. Taylor expanded around 0 3.2

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

3. Final simplification 3.2

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

Reproduce

herbie shell --seed 2019350 +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))))