Average Error: 52.3 → 51.8
Time: 1.1m
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \left(\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}\right)}}\right) + \left(-b\right))_*}{2}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \left(\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}\right)}}\right) + \left(-b\right))_*}{2}}{a}
double f(double a, double b, double c) {
        double r2455896 = b;
        double r2455897 = -r2455896;
        double r2455898 = r2455896 * r2455896;
        double r2455899 = 4.0;
        double r2455900 = a;
        double r2455901 = r2455899 * r2455900;
        double r2455902 = c;
        double r2455903 = r2455901 * r2455902;
        double r2455904 = r2455898 - r2455903;
        double r2455905 = sqrt(r2455904);
        double r2455906 = r2455897 + r2455905;
        double r2455907 = 2.0;
        double r2455908 = r2455907 * r2455900;
        double r2455909 = r2455906 / r2455908;
        return r2455909;
}
↓
double f(double a, double b, double c) {
        double r2455910 = c;
        double r2455911 = a;
        double r2455912 = -4.0;
        double r2455913 = r2455911 * r2455912;
        double r2455914 = b;
        double r2455915 = r2455914 * r2455914;
        double r2455916 = fma(r2455910, r2455913, r2455915);
        double r2455917 = sqrt(r2455916);
        double r2455918 = sqrt(r2455917);
        double r2455919 = cbrt(r2455916);
        double r2455920 = r2455919 * r2455919;
        double r2455921 = r2455919 * r2455920;
        double r2455922 = sqrt(r2455921);
        double r2455923 = sqrt(r2455922);
        double r2455924 = -r2455914;
        double r2455925 = fma(r2455918, r2455923, r2455924);
        double r2455926 = 2.0;
        double r2455927 = r2455925 / r2455926;
        double r2455928 = r2455927 / r2455911;
        return r2455928;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.3

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

  2. Simplified52.3

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} - b}{2}}{a}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt52.3

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

  5. Applied sqrt-prod52.1

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

  6. Applied fma-neg51.5

    \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}}{2}}{a}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt51.8

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) \cdot \sqrt[3]{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}}}\right) + \left(-b\right))_*}{2}}{a}\]

  9. Final simplification51.8

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \left(\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}\right)}}\right) + \left(-b\right))_*}{2}}{a}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))