expq3 (problem 3.4.2)

Average Error: 58.3 → 3.6

Time: 36.7s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.726050140786423e-205
  2. b = 9.786110833240265e+222
  3. eps = 1.1164592442605668e-82

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

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 r2674802 = eps;
        double r2674803 = a;
        double r2674804 = b;
        double r2674805 = r2674803 + r2674804;
        double r2674806 = r2674805 * r2674802;
        double r2674807 = exp(r2674806);
        double r2674808 = 1.0;
        double r2674809 = r2674807 - r2674808;
        double r2674810 = r2674802 * r2674809;
        double r2674811 = r2674803 * r2674802;
        double r2674812 = exp(r2674811);
        double r2674813 = r2674812 - r2674808;
        double r2674814 = r2674804 * r2674802;
        double r2674815 = exp(r2674814);
        double r2674816 = r2674815 - r2674808;
        double r2674817 = r2674813 * r2674816;
        double r2674818 = r2674810 / r2674817;
        return r2674818;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r2674819 = 1.0;
        double r2674820 = a;
        double r2674821 = r2674819 / r2674820;
        double r2674822 = b;
        double r2674823 = r2674819 / r2674822;
        double r2674824 = r2674821 + r2674823;
        return r2674824;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.3
Target	14.5
Herbie	3.6

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

Derivation

1. Initial program 58.3

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

2. Taylor expanded around 0 3.6

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

3. Final simplification 3.6

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

Reproduce

herbie shell --seed 2019133 
(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))))