expq3 (problem 3.4.2)

Average Error: 60.3 → 3.4

Time: 1.1m

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 r41331 = eps;
        double r41332 = a;
        double r41333 = b;
        double r41334 = r41332 + r41333;
        double r41335 = r41334 * r41331;
        double r41336 = exp(r41335);
        double r41337 = 1.0;
        double r41338 = r41336 - r41337;
        double r41339 = r41331 * r41338;
        double r41340 = r41332 * r41331;
        double r41341 = exp(r41340);
        double r41342 = r41341 - r41337;
        double r41343 = r41333 * r41331;
        double r41344 = exp(r41343);
        double r41345 = r41344 - r41337;
        double r41346 = r41342 * r41345;
        double r41347 = r41339 / r41346;
        return r41347;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r41348 = 1.0;
        double r41349 = b;
        double r41350 = r41348 / r41349;
        double r41351 = a;
        double r41352 = r41348 / r41351;
        double r41353 = r41350 + r41352;
        return r41353;
}

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 
(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))))