Average Error: 1.7 → 1.7
Time: 23.1s
Precision: 64
\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]
↓
\[\frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), c, a\right)\right)} - b_2}{a}\]
\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{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), c, a\right)\right)} - b_2}{a}
double f(double a, double b_2, double c) {
        double r864794 = b_2;
        double r864795 = -r864794;
        double r864796 = r864794 * r864794;
        double r864797 = a;
        double r864798 = c;
        double r864799 = r864797 * r864798;
        double r864800 = r864796 - r864799;
        double r864801 = sqrt(r864800);
        double r864802 = r864795 + r864801;
        double r864803 = r864802 / r864797;
        return r864803;
}
↓
double f(double a, double b_2, double c) {
        double r864804 = b_2;
        double r864805 = r864804 * r864804;
        double r864806 = /*Error: no posit support in C */;
        double r864807 = c;
        double r864808 = a;
        double r864809 = /*Error: no posit support in C */;
        double r864810 = /*Error: no posit support in C */;
        double r864811 = sqrt(r864810);
        double r864812 = r864811 - r864804;
        double r864813 = r864812 / r864808;
        return r864813;
}

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. Using strategy rm

  4. Applied introduce-quire1.7

    \[\leadsto \frac{\left(\left(\sqrt{\left(\color{blue}{\left(\left(\left(b_2 \cdot b_2\right)\right)\right)} - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}\]

  5. Applied insert-quire-fdp-sub1.7

    \[\leadsto \frac{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), c, a\right)\right)\right)}}\right) - b_2\right)}{a}\]

  6. Final simplification1.7

    \[\leadsto \frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b_2 \cdot b_2\right)\right), c, a\right)\right)} - b_2}{a}\]

Reproduce

herbie shell --seed 2019163 +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))