Average Error: 1.7 → 1.7
Time: 15.5s
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 r989045 = b_2;
        double r989046 = -r989045;
        double r989047 = r989045 * r989045;
        double r989048 = a;
        double r989049 = c;
        double r989050 = r989048 * r989049;
        double r989051 = r989047 - r989050;
        double r989052 = sqrt(r989051);
        double r989053 = r989046 + r989052;
        double r989054 = r989053 / r989048;
        return r989054;
}
↓
double f(double a, double b_2, double c) {
        double r989055 = b_2;
        double r989056 = r989055 * r989055;
        double r989057 = c;
        double r989058 = a;
        double r989059 = r989057 * r989058;
        double r989060 = r989056 - r989059;
        double r989061 = sqrt(r989060);
        double r989062 = r989061 - r989055;
        double r989063 = r989062 / r989058;
        return r989063;
}

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 2019119 
(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))