Average Error: 1.7 → 1.7
Time: 23.1s
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 r1987365 = b_2;
        double r1987366 = -r1987365;
        double r1987367 = r1987365 * r1987365;
        double r1987368 = a;
        double r1987369 = c;
        double r1987370 = r1987368 * r1987369;
        double r1987371 = r1987367 - r1987370;
        double r1987372 = sqrt(r1987371);
        double r1987373 = r1987366 + r1987372;
        double r1987374 = r1987373 / r1987368;
        return r1987374;
}
↓
double f(double a, double b_2, double c) {
        double r1987375 = b_2;
        double r1987376 = r1987375 * r1987375;
        double r1987377 = c;
        double r1987378 = a;
        double r1987379 = r1987377 * r1987378;
        double r1987380 = r1987376 - r1987379;
        double r1987381 = sqrt(r1987380);
        double r1987382 = r1987381 - r1987375;
        double r1987383 = r1987382 / r1987378;
        return r1987383;
}

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 2019128 +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))