expq3 (problem 3.4.2)

Average Error: 58.9 → 3.1

Time: 46.8s

Precision: 64

• could not determine a ground truth for program body

  1. a = -4.0881825237139195e+193
  2. b = 5.03993005917793e+140
  3. eps = -7.699076133362018e-180

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

Math

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

↓

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

C

double f(double a, double b, double eps) {
        double r3452664 = eps;
        double r3452665 = a;
        double r3452666 = b;
        double r3452667 = r3452665 + r3452666;
        double r3452668 = r3452667 * r3452664;
        double r3452669 = exp(r3452668);
        double r3452670 = 1.0;
        double r3452671 = r3452669 - r3452670;
        double r3452672 = r3452664 * r3452671;
        double r3452673 = r3452665 * r3452664;
        double r3452674 = exp(r3452673);
        double r3452675 = r3452674 - r3452670;
        double r3452676 = r3452666 * r3452664;
        double r3452677 = exp(r3452676);
        double r3452678 = r3452677 - r3452670;
        double r3452679 = r3452675 * r3452678;
        double r3452680 = r3452672 / r3452679;
        return r3452680;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r3452681 = 1.0;
        double r3452682 = a;
        double r3452683 = r3452681 / r3452682;
        double r3452684 = b;
        double r3452685 = r3452681 / r3452684;
        double r3452686 = r3452683 + r3452685;
        return r3452686;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.9
Target	14.1
Herbie	3.1

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

Derivation

1. Initial program 58.9

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

2. Taylor expanded around 0 3.1

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

3. Final simplification 3.1

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

Reproduce

herbie shell --seed 2019151 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))