Average Error: 34.0 → 9.1
Time: 19.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\

\mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r5479883 = b;
        double r5479884 = -r5479883;
        double r5479885 = r5479883 * r5479883;
        double r5479886 = 4.0;
        double r5479887 = a;
        double r5479888 = r5479886 * r5479887;
        double r5479889 = c;
        double r5479890 = r5479888 * r5479889;
        double r5479891 = r5479885 - r5479890;
        double r5479892 = sqrt(r5479891);
        double r5479893 = r5479884 + r5479892;
        double r5479894 = 2.0;
        double r5479895 = r5479894 * r5479887;
        double r5479896 = r5479893 / r5479895;
        return r5479896;
}
↓
double f(double a, double b, double c) {
        double r5479897 = b;
        double r5479898 = -2.900769547116861e+46;
        bool r5479899 = r5479897 <= r5479898;
        double r5479900 = c;
        double r5479901 = r5479900 / r5479897;
        double r5479902 = a;
        double r5479903 = r5479897 / r5479902;
        double r5479904 = r5479901 - r5479903;
        double r5479905 = 2.0;
        double r5479906 = r5479904 * r5479905;
        double r5479907 = r5479906 / r5479905;
        double r5479908 = 5.4213851798811764e-102;
        bool r5479909 = r5479897 <= r5479908;
        double r5479910 = 1.0;
        double r5479911 = r5479910 / r5479902;
        double r5479912 = -4.0;
        double r5479913 = r5479912 * r5479902;
        double r5479914 = r5479913 * r5479900;
        double r5479915 = fma(r5479897, r5479897, r5479914);
        double r5479916 = sqrt(r5479915);
        double r5479917 = r5479916 - r5479897;
        double r5479918 = r5479911 * r5479917;
        double r5479919 = r5479918 / r5479905;
        double r5479920 = 1.1597179970514171e+23;
        bool r5479921 = r5479897 <= r5479920;
        double r5479922 = cbrt(r5479902);
        double r5479923 = r5479922 * r5479922;
        double r5479924 = r5479910 / r5479923;
        double r5479925 = 0.0;
        double r5479926 = fma(r5479900, r5479913, r5479925);
        double r5479927 = r5479916 + r5479897;
        double r5479928 = r5479926 / r5479927;
        double r5479929 = r5479928 / r5479922;
        double r5479930 = r5479924 * r5479929;
        double r5479931 = r5479930 / r5479905;
        double r5479932 = -2.0;
        double r5479933 = r5479932 * r5479901;
        double r5479934 = r5479933 / r5479905;
        double r5479935 = r5479921 ? r5479931 : r5479934;
        double r5479936 = r5479909 ? r5479919 : r5479935;
        double r5479937 = r5479899 ? r5479907 : r5479936;
        return r5479937;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.900769547116861e+46

    1. Initial program 35.9

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

    2. Simplified35.9

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

    4. Applied div-inv36.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{1}{a}}}{2}\]

    5. Taylor expanded around -inf 5.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    6. Simplified5.3

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

    if -2.900769547116861e+46 < b < 5.4213851798811764e-102

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied div-inv12.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{1}{a}}}{2}\]

    if 5.4213851798811764e-102 < b < 1.1597179970514171e+23

    1. Initial program 39.1

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

    2. Simplified39.1

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

    4. Applied add-cube-cbrt39.5

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}}{2}\]

    5. Applied *-un-lft-identity39.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{2}\]

    6. Applied times-frac39.5

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{\sqrt[3]{a}}}}{2}\]
    7. Using strategy rm

    8. Applied flip--39.6

      \[\leadsto \frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}}{\sqrt[3]{a}}}{2}\]

    9. Simplified17.8

      \[\leadsto \frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\color{blue}{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\]

    if 1.1597179970514171e+23 < b

    1. Initial program 55.6

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

    2. Simplified55.6

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

    3. Taylor expanded around inf 4.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.900769547116861 \cdot 10^{+46}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.4213851798811764 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.1597179970514171 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, 0\right)}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} + b}}{\sqrt[3]{a}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))