expq3 (problem 3.4.2)

Average Error: 60.1 → 3.6

Time: 16.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 5.147318504951488e+114
  2. b = -2.4549914704659062e+268
  3. eps = -7.793413198338314e-36

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

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 r373154 = eps;
        double r373155 = a;
        double r373156 = b;
        double r373157 = r373155 + r373156;
        double r373158 = r373157 * r373154;
        double r373159 = exp(r373158);
        double r373160 = 1.0;
        double r373161 = r373159 - r373160;
        double r373162 = r373154 * r373161;
        double r373163 = r373155 * r373154;
        double r373164 = exp(r373163);
        double r373165 = r373164 - r373160;
        double r373166 = r373156 * r373154;
        double r373167 = exp(r373166);
        double r373168 = r373167 - r373160;
        double r373169 = r373165 * r373168;
        double r373170 = r373162 / r373169;
        return r373170;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r373171 = 1.0;
        double r373172 = b;
        double r373173 = r373171 / r373172;
        double r373174 = a;
        double r373175 = r373171 / r373174;
        double r373176 = r373173 + r373175;
        return r373176;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.1
Target	14.8
Herbie	3.6

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

Derivation

1. Initial program 60.1

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

2. Taylor expanded around 0 3.6

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

3. Final simplification 3.6

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

Reproduce

herbie shell --seed 350497007 
(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))))