expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 35.6s

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 r87314 = eps;
        double r87315 = a;
        double r87316 = b;
        double r87317 = r87315 + r87316;
        double r87318 = r87317 * r87314;
        double r87319 = exp(r87318);
        double r87320 = 1.0;
        double r87321 = r87319 - r87320;
        double r87322 = r87314 * r87321;
        double r87323 = r87315 * r87314;
        double r87324 = exp(r87323);
        double r87325 = r87324 - r87320;
        double r87326 = r87316 * r87314;
        double r87327 = exp(r87326);
        double r87328 = r87327 - r87320;
        double r87329 = r87325 * r87328;
        double r87330 = r87322 / r87329;
        return r87330;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r87331 = 1.0;
        double r87332 = b;
        double r87333 = r87331 / r87332;
        double r87334 = a;
        double r87335 = r87331 / r87334;
        double r87336 = r87333 + r87335;
        return r87336;
}

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))))