Average Error: 1.7 → 1.7
Time: 3.2m
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\]
\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}
double f(double a, double b_2, double c) {
        double r1184776 = b_2;
        double r1184777 = -r1184776;
        double r1184778 = r1184776 * r1184776;
        double r1184779 = a;
        double r1184780 = c;
        double r1184781 = r1184779 * r1184780;
        double r1184782 = r1184778 - r1184781;
        double r1184783 = sqrt(r1184782);
        double r1184784 = r1184777 + r1184783;
        double r1184785 = r1184784 / r1184779;
        return r1184785;
}
↓
double f(double a, double b_2, double c) {
        double r1184786 = b_2;
        double r1184787 = r1184786 * r1184786;
        double r1184788 = c;
        double r1184789 = a;
        double r1184790 = r1184788 * r1184789;
        double r1184791 = r1184787 - r1184790;
        double r1184792 = sqrt(r1184791);
        double r1184793 = r1184792 - r1184786;
        double r1184794 = r1184793 / r1184789;
        return r1184794;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.7

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

  2. Simplified1.7

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

  3. Final simplification1.7

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

Reproduce

herbie shell --seed 2019152 +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))