Average Error: 1.5 → 1.5
Time: 16.9s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}\]
\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)}
↓
\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2}}{a}
double f(double a, double b, double c) {
        double r674516 = b;
        double r674517 = -r674516;
        double r674518 = r674516 * r674516;
        double r674519 = 4.0;
        double r674520 = /* ERROR: no posit support in C */;
        double r674521 = a;
        double r674522 = c;
        double r674523 = r674521 * r674522;
        double r674524 = r674520 * r674523;
        double r674525 = r674518 - r674524;
        double r674526 = sqrt(r674525);
        double r674527 = r674517 + r674526;
        double r674528 = 2.0;
        double r674529 = /* ERROR: no posit support in C */;
        double r674530 = r674529 * r674521;
        double r674531 = r674527 / r674530;
        return r674531;
}
↓
double f(double a, double b, double c) {
        double r674532 = b;
        double r674533 = r674532 * r674532;
        double r674534 = c;
        double r674535 = a;
        double r674536 = r674534 * r674535;
        double r674537 = 4.0;
        double r674538 = r674536 * r674537;
        double r674539 = r674533 - r674538;
        double r674540 = sqrt(r674539);
        double r674541 = r674540 - r674532;
        double r674542 = 2.0;
        double r674543 = r674541 / r674542;
        double r674544 = r674543 / r674535;
        return r674544;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied associate-/r*1.5

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

  5. Final simplification1.5

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

Reproduce

herbie shell --seed 2019124 
(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)))