Average Error: 34.9 → 14.1
Time: 19.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.486593375467686151654921844207164570885 \cdot 10^{143}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.198567300310805542976550787106877171711 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]
\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 -2.486593375467686151654921844207164570885 \cdot 10^{143}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.198567300310805542976550787106877171711 \cdot 10^{-33}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r91847 = b;
        double r91848 = -r91847;
        double r91849 = r91847 * r91847;
        double r91850 = 4.0;
        double r91851 = a;
        double r91852 = r91850 * r91851;
        double r91853 = c;
        double r91854 = r91852 * r91853;
        double r91855 = r91849 - r91854;
        double r91856 = sqrt(r91855);
        double r91857 = r91848 + r91856;
        double r91858 = 2.0;
        double r91859 = r91858 * r91851;
        double r91860 = r91857 / r91859;
        return r91860;
}
↓
double f(double a, double b, double c) {
        double r91861 = b;
        double r91862 = -2.486593375467686e+143;
        bool r91863 = r91861 <= r91862;
        double r91864 = -2.0;
        double r91865 = r91864 * r91861;
        double r91866 = 2.0;
        double r91867 = a;
        double r91868 = r91866 * r91867;
        double r91869 = r91865 / r91868;
        double r91870 = 5.1985673003108055e-33;
        bool r91871 = r91861 <= r91870;
        double r91872 = r91861 * r91861;
        double r91873 = 4.0;
        double r91874 = r91873 * r91867;
        double r91875 = c;
        double r91876 = r91874 * r91875;
        double r91877 = r91872 - r91876;
        double r91878 = sqrt(r91877);
        double r91879 = r91878 - r91861;
        double r91880 = r91879 / r91868;
        double r91881 = -2.0;
        double r91882 = r91867 * r91875;
        double r91883 = r91882 / r91861;
        double r91884 = r91881 * r91883;
        double r91885 = r91884 / r91868;
        double r91886 = r91871 ? r91880 : r91885;
        double r91887 = r91863 ? r91869 : r91886;
        return r91887;
}

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

Original34.9
Target21.3
Herbie14.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.486593375467686e+143

    1. Initial program 59.5

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

    2. Simplified59.5

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

    4. Applied add-cube-cbrt59.5

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

    5. Applied sqrt-prod59.5

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

    6. Applied fma-neg59.6

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

    7. Taylor expanded around -inf 2.7

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

    8. Simplified2.7

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

    if -2.486593375467686e+143 < b < 5.1985673003108055e-33

    1. Initial program 14.5

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

    2. Simplified14.5

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

    if 5.1985673003108055e-33 < b

    1. Initial program 55.5

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

    2. Simplified55.5

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

    3. Taylor expanded around inf 17.3

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.486593375467686151654921844207164570885 \cdot 10^{143}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.198567300310805542976550787106877171711 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{a \cdot c}{b}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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)))