expq3 (problem 3.4.2)

Average Error: 60.2 → 3.4

Time: 10.8s

Precision: 64

• could not determine a ground truth for program body

  1. a = -2.147009436181225e-259
  2. b = 1.0551194375545526e+290
  3. eps = 3.8333244974596404e-56

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

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 r74086 = eps;
        double r74087 = a;
        double r74088 = b;
        double r74089 = r74087 + r74088;
        double r74090 = r74089 * r74086;
        double r74091 = exp(r74090);
        double r74092 = 1.0;
        double r74093 = r74091 - r74092;
        double r74094 = r74086 * r74093;
        double r74095 = r74087 * r74086;
        double r74096 = exp(r74095);
        double r74097 = r74096 - r74092;
        double r74098 = r74088 * r74086;
        double r74099 = exp(r74098);
        double r74100 = r74099 - r74092;
        double r74101 = r74097 * r74100;
        double r74102 = r74094 / r74101;
        return r74102;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r74103 = 1.0;
        double r74104 = b;
        double r74105 = r74103 / r74104;
        double r74106 = a;
        double r74107 = r74103 / r74106;
        double r74108 = r74105 + r74107;
        return r74108;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	14.7
Herbie	3.4

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

Derivation

1. Initial program 60.2

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

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 2020060 
(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))))