Average Error: 34.2 → 11.9
Time: 15.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\ \;\;\;\;\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \frac{\sqrt[3]{c}}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\
\;\;\;\;\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \frac{\sqrt[3]{c}}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r118906 = b;
        double r118907 = -r118906;
        double r118908 = r118906 * r118906;
        double r118909 = 4.0;
        double r118910 = a;
        double r118911 = r118909 * r118910;
        double r118912 = c;
        double r118913 = r118911 * r118912;
        double r118914 = r118908 - r118913;
        double r118915 = sqrt(r118914);
        double r118916 = r118907 + r118915;
        double r118917 = 2.0;
        double r118918 = r118917 * r118910;
        double r118919 = r118916 / r118918;
        return r118919;
}
↓
double f(double a, double b, double c) {
        double r118920 = b;
        double r118921 = -1.5476666036365373e+50;
        bool r118922 = r118920 <= r118921;
        double r118923 = c;
        double r118924 = cbrt(r118923);
        double r118925 = r118924 * r118924;
        double r118926 = r118924 / r118920;
        double r118927 = r118925 * r118926;
        double r118928 = a;
        double r118929 = r118920 / r118928;
        double r118930 = r118927 - r118929;
        double r118931 = 1.0;
        double r118932 = r118930 * r118931;
        double r118933 = 7.455592343308264e-170;
        bool r118934 = r118920 <= r118933;
        double r118935 = 1.0;
        double r118936 = 2.0;
        double r118937 = r118936 * r118928;
        double r118938 = r118920 * r118920;
        double r118939 = 4.0;
        double r118940 = r118939 * r118928;
        double r118941 = r118940 * r118923;
        double r118942 = r118938 - r118941;
        double r118943 = sqrt(r118942);
        double r118944 = r118943 - r118920;
        double r118945 = r118937 / r118944;
        double r118946 = r118935 / r118945;
        double r118947 = -1.0;
        double r118948 = r118923 / r118920;
        double r118949 = r118947 * r118948;
        double r118950 = r118934 ? r118946 : r118949;
        double r118951 = r118922 ? r118932 : r118950;
        return r118951;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.2
Target20.8
Herbie11.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5476666036365373e+50

    1. Initial program 37.8

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

    2. Simplified37.8

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

    3. Taylor expanded around -inf 5.8

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    4. Simplified5.8

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.0

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

    7. Applied *-un-lft-identity7.0

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

    8. Applied add-cube-cbrt7.0

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

    9. Applied times-frac7.0

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

    10. Applied prod-diff7.0

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

    11. Simplified5.8

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

    12. Simplified5.8

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

    if -1.5476666036365373e+50 < b < 7.455592343308264e-170

    1. Initial program 12.4

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

    2. Simplified12.4

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

    4. Applied clear-num12.5

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

    if 7.455592343308264e-170 < b

    1. Initial program 48.9

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

    2. Simplified48.9

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

    3. Taylor expanded around inf 14.1

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.547666603636537260513437138645901028344 \cdot 10^{50}:\\ \;\;\;\;\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \frac{\sqrt[3]{c}}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))