Average Error: 1.7 → 1.7
Time: 23.2s
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 r282592 = b_2;
        double r282593 = -r282592;
        double r282594 = r282592 * r282592;
        double r282595 = a;
        double r282596 = c;
        double r282597 = r282595 * r282596;
        double r282598 = r282594 - r282597;
        double r282599 = sqrt(r282598);
        double r282600 = r282593 + r282599;
        double r282601 = r282600 / r282595;
        return r282601;
}
↓
double f(double a, double b_2, double c) {
        double r282602 = b_2;
        double r282603 = r282602 * r282602;
        double r282604 = c;
        double r282605 = a;
        double r282606 = r282604 * r282605;
        double r282607 = r282603 - r282606;
        double r282608 = sqrt(r282607);
        double r282609 = r282608 - r282602;
        double r282610 = r282609 / r282605;
        return r282610;
}

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 2019152 +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))