Average Error: 1.5 → 1.5
Time: 34.1s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}
↓
\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
double f(double a, double b, double c) {
        double r1094010 = b;
        double r1094011 = -r1094010;
        double r1094012 = r1094010 * r1094010;
        double r1094013 = 4.0;
        double r1094014 = /* ERROR: no posit support in C */;
        double r1094015 = a;
        double r1094016 = c;
        double r1094017 = r1094015 * r1094016;
        double r1094018 = r1094014 * r1094017;
        double r1094019 = r1094012 - r1094018;
        double r1094020 = sqrt(r1094019);
        double r1094021 = r1094011 - r1094020;
        double r1094022 = 2.0;
        double r1094023 = /* ERROR: no posit support in C */;
        double r1094024 = r1094023 * r1094015;
        double r1094025 = r1094021 / r1094024;
        return r1094025;
}
↓
double f(double a, double b, double c) {
        double r1094026 = b;
        double r1094027 = -r1094026;
        double r1094028 = r1094026 * r1094026;
        double r1094029 = 4.0;
        double r1094030 = a;
        double r1094031 = r1094029 * r1094030;
        double r1094032 = c;
        double r1094033 = r1094031 * r1094032;
        double r1094034 = r1094028 - r1094033;
        double r1094035 = sqrt(r1094034);
        double r1094036 = r1094027 - r1094035;
        double r1094037 = 2.0;
        double r1094038 = r1094037 * r1094030;
        double r1094039 = r1094036 / r1094038;
        return r1094039;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

    \[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
  2. Using strategy rm

  3. Applied associate-*r*1.5

    \[\leadsto \frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \color{blue}{\left(\left(\left(4\right) \cdot a\right) \cdot c\right)}\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

  4. Final simplification1.5

    \[\leadsto \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))