expq3 (problem 3.4.2)

Average Error: 58.9 → 3.1

Time: 30.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = -4.0881825237139195e+193
  2. b = 5.03993005917793e+140
  3. eps = -7.699076133362018e-180

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

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 r2132195 = eps;
        double r2132196 = a;
        double r2132197 = b;
        double r2132198 = r2132196 + r2132197;
        double r2132199 = r2132198 * r2132195;
        double r2132200 = exp(r2132199);
        double r2132201 = 1.0;
        double r2132202 = r2132200 - r2132201;
        double r2132203 = r2132195 * r2132202;
        double r2132204 = r2132196 * r2132195;
        double r2132205 = exp(r2132204);
        double r2132206 = r2132205 - r2132201;
        double r2132207 = r2132197 * r2132195;
        double r2132208 = exp(r2132207);
        double r2132209 = r2132208 - r2132201;
        double r2132210 = r2132206 * r2132209;
        double r2132211 = r2132203 / r2132210;
        return r2132211;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r2132212 = 1.0;
        double r2132213 = a;
        double r2132214 = r2132212 / r2132213;
        double r2132215 = b;
        double r2132216 = r2132212 / r2132215;
        double r2132217 = r2132214 + r2132216;
        return r2132217;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.9
Target	14.1
Herbie	3.1

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

Derivation

1. Initial program 58.9

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

2. Taylor expanded around 0 3.1

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

3. Final simplification 3.1

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

Reproduce

herbie shell --seed 2019151 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))