Average Error: 1.7 → 1.7
Time: 16.9s
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 r1179746 = b_2;
        double r1179747 = -r1179746;
        double r1179748 = r1179746 * r1179746;
        double r1179749 = a;
        double r1179750 = c;
        double r1179751 = r1179749 * r1179750;
        double r1179752 = r1179748 - r1179751;
        double r1179753 = sqrt(r1179752);
        double r1179754 = r1179747 + r1179753;
        double r1179755 = r1179754 / r1179749;
        return r1179755;
}
↓
double f(double a, double b_2, double c) {
        double r1179756 = b_2;
        double r1179757 = r1179756 * r1179756;
        double r1179758 = c;
        double r1179759 = a;
        double r1179760 = r1179758 * r1179759;
        double r1179761 = r1179757 - r1179760;
        double r1179762 = sqrt(r1179761);
        double r1179763 = r1179762 - r1179756;
        double r1179764 = r1179763 / r1179759;
        return r1179764;
}

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 2019133 +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))