expq3 (problem 3.4.2)

Average Error: 60.4 → 3.3

Time: 19.4s

Precision: 64

• could not determine a ground truth for program body

  1. a = 2.8774590365534544e-144
  2. b = 3.0260304830237114e+184
  3. eps = 1.0722929270979165e-66

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

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 r108987 = eps;
        double r108988 = a;
        double r108989 = b;
        double r108990 = r108988 + r108989;
        double r108991 = r108990 * r108987;
        double r108992 = exp(r108991);
        double r108993 = 1.0;
        double r108994 = r108992 - r108993;
        double r108995 = r108987 * r108994;
        double r108996 = r108988 * r108987;
        double r108997 = exp(r108996);
        double r108998 = r108997 - r108993;
        double r108999 = r108989 * r108987;
        double r109000 = exp(r108999);
        double r109001 = r109000 - r108993;
        double r109002 = r108998 * r109001;
        double r109003 = r108995 / r109002;
        return r109003;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r109004 = 1.0;
        double r109005 = b;
        double r109006 = r109004 / r109005;
        double r109007 = a;
        double r109008 = r109004 / r109007;
        double r109009 = r109006 + r109008;
        return r109009;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.8
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 3.3

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

3. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019304 
(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))))