expq3 (problem 3.4.2)

Average Error: 58.7 → 3.3

Time: 32.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = -7.04231306472785e-44
  2. b = 1.461159083214139e+105
  3. eps = 2.6320811315441853e-51

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

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 r4772044 = eps;
        double r4772045 = a;
        double r4772046 = b;
        double r4772047 = r4772045 + r4772046;
        double r4772048 = r4772047 * r4772044;
        double r4772049 = exp(r4772048);
        double r4772050 = 1.0;
        double r4772051 = r4772049 - r4772050;
        double r4772052 = r4772044 * r4772051;
        double r4772053 = r4772045 * r4772044;
        double r4772054 = exp(r4772053);
        double r4772055 = r4772054 - r4772050;
        double r4772056 = r4772046 * r4772044;
        double r4772057 = exp(r4772056);
        double r4772058 = r4772057 - r4772050;
        double r4772059 = r4772055 * r4772058;
        double r4772060 = r4772052 / r4772059;
        return r4772060;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r4772061 = 1.0;
        double r4772062 = a;
        double r4772063 = r4772061 / r4772062;
        double r4772064 = b;
        double r4772065 = r4772061 / r4772064;
        double r4772066 = r4772063 + r4772065;
        return r4772066;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.7
Target	14.5
Herbie	3.3

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

Derivation

1. Initial program 58.7

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

2. Taylor expanded around 0 3.3

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

3. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019162 
(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))))