expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 33.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = -5.278031834180875e+72
  2. b = -5.9461326997729256e+41
  3. eps = -2.3299265771866867e-17

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

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 r28960 = eps;
        double r28961 = a;
        double r28962 = b;
        double r28963 = r28961 + r28962;
        double r28964 = r28963 * r28960;
        double r28965 = exp(r28964);
        double r28966 = 1.0;
        double r28967 = r28965 - r28966;
        double r28968 = r28960 * r28967;
        double r28969 = r28961 * r28960;
        double r28970 = exp(r28969);
        double r28971 = r28970 - r28966;
        double r28972 = r28962 * r28960;
        double r28973 = exp(r28972);
        double r28974 = r28973 - r28966;
        double r28975 = r28971 * r28974;
        double r28976 = r28968 / r28975;
        return r28976;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r28977 = 1.0;
        double r28978 = b;
        double r28979 = r28977 / r28978;
        double r28980 = a;
        double r28981 = r28977 / r28980;
        double r28982 = r28979 + r28981;
        return r28982;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.2
Herbie	3.4

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

Derivation

1. Initial program 60.3

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

2. Taylor expanded around 0 3.4

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

3. Final simplification 3.4

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))