expq3 (problem 3.4.2)

Average Error: 60.5 → 3.2

Time: 16.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = 421651.551737869
  2. b = 8.638709115370421e+217
  3. eps = 3.122426261387777e-55

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

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 r54527 = eps;
        double r54528 = a;
        double r54529 = b;
        double r54530 = r54528 + r54529;
        double r54531 = r54530 * r54527;
        double r54532 = exp(r54531);
        double r54533 = 1.0;
        double r54534 = r54532 - r54533;
        double r54535 = r54527 * r54534;
        double r54536 = r54528 * r54527;
        double r54537 = exp(r54536);
        double r54538 = r54537 - r54533;
        double r54539 = r54529 * r54527;
        double r54540 = exp(r54539);
        double r54541 = r54540 - r54533;
        double r54542 = r54538 * r54541;
        double r54543 = r54535 / r54542;
        return r54543;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r54544 = 1.0;
        double r54545 = b;
        double r54546 = r54544 / r54545;
        double r54547 = a;
        double r54548 = r54544 / r54547;
        double r54549 = r54546 + r54548;
        return r54549;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.5
Target	15.0
Herbie	3.2

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

Derivation

1. Initial program 60.5

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

2. Taylor expanded around 0 3.2

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

3. Final simplification 3.2

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

Reproduce

herbie shell --seed 197574269 
(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))))