Average Error: 1.8 → 1.8
Time: 20.5s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
double f(double a, double b_2, double c) {
        double r1089021 = b_2;
        double r1089022 = -r1089021;
        double r1089023 = r1089021 * r1089021;
        double r1089024 = a;
        double r1089025 = c;
        double r1089026 = r1089024 * r1089025;
        double r1089027 = r1089023 - r1089026;
        double r1089028 = sqrt(r1089027);
        double r1089029 = r1089022 - r1089028;
        double r1089030 = r1089029 / r1089024;
        return r1089030;
}
↓
double f(double a, double b_2, double c) {
        double r1089031 = b_2;
        double r1089032 = -r1089031;
        double r1089033 = r1089031 * r1089031;
        double r1089034 = a;
        double r1089035 = c;
        double r1089036 = r1089034 * r1089035;
        double r1089037 = r1089033 - r1089036;
        double r1089038 = sqrt(r1089037);
        double r1089039 = r1089032 - r1089038;
        double r1089040 = r1089039 / r1089034;
        return r1089040;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Initial program 1.8

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

  2. Final simplification1.8

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

Reproduce

herbie shell --seed 2019165 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))