Average Error: 1.6 → 1.6
Time: 19.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
double f(double a, double b, double c) {
        double r788399 = b;
        double r788400 = -r788399;
        double r788401 = r788399 * r788399;
        double r788402 = 4.0;
        double r788403 = a;
        double r788404 = c;
        double r788405 = r788403 * r788404;
        double r788406 = r788402 * r788405;
        double r788407 = r788401 - r788406;
        double r788408 = sqrt(r788407);
        double r788409 = r788400 + r788408;
        double r788410 = 2.0;
        double r788411 = r788410 * r788403;
        double r788412 = r788409 / r788411;
        return r788412;
}
↓
double f(double a, double b, double c) {
        double r788413 = b;
        double r788414 = r788413 * r788413;
        double r788415 = c;
        double r788416 = a;
        double r788417 = r788415 * r788416;
        double r788418 = 4.0;
        double r788419 = r788417 * r788418;
        double r788420 = r788414 - r788419;
        double r788421 = sqrt(r788420);
        double r788422 = r788421 - r788413;
        double r788423 = 2.0;
        double r788424 = r788423 * r788416;
        double r788425 = r788422 / r788424;
        return r788425;
}


\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

    \[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

  2. Simplified1.6

    \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(\left(2\right) \cdot a\right)}}\]

  3. Final simplification1.6

    \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))