Average Error: 1.7 → 1.7
Time: 15.1s
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 r1010744 = b_2;
        double r1010745 = -r1010744;
        double r1010746 = r1010744 * r1010744;
        double r1010747 = a;
        double r1010748 = c;
        double r1010749 = r1010747 * r1010748;
        double r1010750 = r1010746 - r1010749;
        double r1010751 = sqrt(r1010750);
        double r1010752 = r1010745 + r1010751;
        double r1010753 = r1010752 / r1010747;
        return r1010753;
}
↓
double f(double a, double b_2, double c) {
        double r1010754 = b_2;
        double r1010755 = r1010754 * r1010754;
        double r1010756 = c;
        double r1010757 = a;
        double r1010758 = r1010756 * r1010757;
        double r1010759 = r1010755 - r1010758;
        double r1010760 = sqrt(r1010759);
        double r1010761 = r1010760 - r1010754;
        double r1010762 = r1010761 / r1010757;
        return r1010762;
}

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 2019124 
(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))