Average Error: 1.7 → 1.7
Time: 25.9s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
double f(double a, double b_2, double c) {
        double r967774 = b_2;
        double r967775 = -r967774;
        double r967776 = r967774 * r967774;
        double r967777 = a;
        double r967778 = c;
        double r967779 = r967777 * r967778;
        double r967780 = r967776 - r967779;
        double r967781 = sqrt(r967780);
        double r967782 = r967775 - r967781;
        double r967783 = r967782 / r967777;
        return r967783;
}
↓
double f(double a, double b_2, double c) {
        double r967784 = b_2;
        double r967785 = -r967784;
        double r967786 = r967784 * r967784;
        double r967787 = a;
        double r967788 = c;
        double r967789 = r967787 * r967788;
        double r967790 = r967786 - r967789;
        double r967791 = sqrt(r967790);
        double r967792 = r967785 - r967791;
        double r967793 = r967792 / r967787;
        return r967793;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

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

  2. Final simplification1.7

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

Reproduce

herbie shell --seed 2019162 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))