quad2p (problem 3.2.1, positive)

Average Error: 1.6 → 1.6

Time: 16.8s

Precision: 64

Math

\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

↓

\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]

C

double f(double a, double b_2, double c) {
        double r1020744 = b_2;
        double r1020745 = -r1020744;
        double r1020746 = r1020744 * r1020744;
        double r1020747 = a;
        double r1020748 = c;
        double r1020749 = r1020747 * r1020748;
        double r1020750 = r1020746 - r1020749;
        double r1020751 = sqrt(r1020750);
        double r1020752 = r1020745 + r1020751;
        double r1020753 = r1020752 / r1020747;
        return r1020753;
}

↓

double f(double a, double b_2, double c) {
        double r1020754 = b_2;
        double r1020755 = r1020754 * r1020754;
        double r1020756 = c;
        double r1020757 = a;
        double r1020758 = r1020756 * r1020757;
        double r1020759 = r1020755 - r1020758;
        double r1020760 = sqrt(r1020759);
        double r1020761 = r1020760 - r1020754;
        double r1020762 = r1020761 / r1020757;
        return r1020762;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Initial program 1.6

   \[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

2. Simplified 1.6

   \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}}\]

3. Final simplification 1.6

   \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/.p16 (+.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))