Average Error: 33.6 → 10.3
Time: 7.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r107832 = b;
        double r107833 = -r107832;
        double r107834 = r107832 * r107832;
        double r107835 = 4.0;
        double r107836 = a;
        double r107837 = c;
        double r107838 = r107836 * r107837;
        double r107839 = r107835 * r107838;
        double r107840 = r107834 - r107839;
        double r107841 = sqrt(r107840);
        double r107842 = r107833 - r107841;
        double r107843 = 2.0;
        double r107844 = r107843 * r107836;
        double r107845 = r107842 / r107844;
        return r107845;
}
↓
double f(double a, double b, double c) {
        double r107846 = b;
        double r107847 = -4.1690865718193236e-104;
        bool r107848 = r107846 <= r107847;
        double r107849 = -1.0;
        double r107850 = c;
        double r107851 = r107850 / r107846;
        double r107852 = r107849 * r107851;
        double r107853 = 1.3316184968738608e+61;
        bool r107854 = r107846 <= r107853;
        double r107855 = 1.0;
        double r107856 = 2.0;
        double r107857 = r107855 / r107856;
        double r107858 = -r107846;
        double r107859 = r107846 * r107846;
        double r107860 = 4.0;
        double r107861 = a;
        double r107862 = r107861 * r107850;
        double r107863 = r107860 * r107862;
        double r107864 = r107859 - r107863;
        double r107865 = sqrt(r107864);
        double r107866 = r107858 - r107865;
        double r107867 = r107866 / r107861;
        double r107868 = r107857 * r107867;
        double r107869 = -2.0;
        double r107870 = r107846 / r107861;
        double r107871 = r107869 * r107870;
        double r107872 = r107857 * r107871;
        double r107873 = r107854 ? r107868 : r107872;
        double r107874 = r107848 ? r107852 : r107873;
        return r107874;
}

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

Original33.6
Target20.6
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.1690865718193236e-104

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 11.0

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

    if -4.1690865718193236e-104 < b < 1.3316184968738608e+61

    1. Initial program 12.3

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

    3. Applied *-un-lft-identity12.3

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

    4. Applied times-frac12.3

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

    6. Applied clear-num12.4

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

    8. Applied *-un-lft-identity12.4

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

    9. Applied *-un-lft-identity12.4

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

    10. Applied times-frac12.4

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

    11. Applied add-cube-cbrt12.4

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

    12. Applied times-frac12.4

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

    13. Simplified12.4

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

    14. Simplified12.3

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

    if 1.3316184968738608e+61 < b

    1. Initial program 39.6

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

    3. Applied *-un-lft-identity39.6

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

    4. Applied times-frac39.5

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

    6. Applied clear-num39.6

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

    7. Taylor expanded around 0 4.6

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(-2 \cdot \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

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