Average Error: 1.7 → 1.7
Time: 21.6s
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 r1980956 = b_2;
        double r1980957 = -r1980956;
        double r1980958 = r1980956 * r1980956;
        double r1980959 = a;
        double r1980960 = c;
        double r1980961 = r1980959 * r1980960;
        double r1980962 = r1980958 - r1980961;
        double r1980963 = sqrt(r1980962);
        double r1980964 = r1980957 + r1980963;
        double r1980965 = r1980964 / r1980959;
        return r1980965;
}
↓
double f(double a, double b_2, double c) {
        double r1980966 = b_2;
        double r1980967 = r1980966 * r1980966;
        double r1980968 = c;
        double r1980969 = a;
        double r1980970 = r1980968 * r1980969;
        double r1980971 = r1980967 - r1980970;
        double r1980972 = sqrt(r1980971);
        double r1980973 = r1980972 - r1980966;
        double r1980974 = r1980973 / r1980969;
        return r1980974;
}

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 2019125 
(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))