Average Error: 1.7 → 1.7
Time: 17.0s
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 r989002 = b_2;
        double r989003 = -r989002;
        double r989004 = r989002 * r989002;
        double r989005 = a;
        double r989006 = c;
        double r989007 = r989005 * r989006;
        double r989008 = r989004 - r989007;
        double r989009 = sqrt(r989008);
        double r989010 = r989003 + r989009;
        double r989011 = r989010 / r989005;
        return r989011;
}
↓
double f(double a, double b_2, double c) {
        double r989012 = b_2;
        double r989013 = r989012 * r989012;
        double r989014 = c;
        double r989015 = a;
        double r989016 = r989014 * r989015;
        double r989017 = r989013 - r989016;
        double r989018 = sqrt(r989017);
        double r989019 = r989018 - r989012;
        double r989020 = r989019 / r989015;
        return r989020;
}

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 2019120 
(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))