Average Error: 33.3 → 6.4
Time: 1.1m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a} \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.263941314600607 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a} \cdot \frac{1}{2}\\

\mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\
\;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r38382886 = b;
        double r38382887 = -r38382886;
        double r38382888 = r38382886 * r38382886;
        double r38382889 = 4.0;
        double r38382890 = a;
        double r38382891 = r38382889 * r38382890;
        double r38382892 = c;
        double r38382893 = r38382891 * r38382892;
        double r38382894 = r38382888 - r38382893;
        double r38382895 = sqrt(r38382894);
        double r38382896 = r38382887 + r38382895;
        double r38382897 = 2.0;
        double r38382898 = r38382897 * r38382890;
        double r38382899 = r38382896 / r38382898;
        return r38382899;
}
↓
double f(double a, double b, double c) {
        double r38382900 = b;
        double r38382901 = -3.263941314600607e+152;
        bool r38382902 = r38382900 <= r38382901;
        double r38382903 = c;
        double r38382904 = r38382903 / r38382900;
        double r38382905 = a;
        double r38382906 = r38382900 / r38382905;
        double r38382907 = r38382904 - r38382906;
        double r38382908 = -4.687918346756617e-254;
        bool r38382909 = r38382900 <= r38382908;
        double r38382910 = r38382900 * r38382900;
        double r38382911 = r38382903 * r38382905;
        double r38382912 = 4.0;
        double r38382913 = r38382911 * r38382912;
        double r38382914 = r38382910 - r38382913;
        double r38382915 = sqrt(r38382914);
        double r38382916 = r38382915 - r38382900;
        double r38382917 = r38382916 / r38382905;
        double r38382918 = 0.5;
        double r38382919 = r38382917 * r38382918;
        double r38382920 = 3.463606471108268e+121;
        bool r38382921 = r38382900 <= r38382920;
        double r38382922 = -2.0;
        double r38382923 = r38382903 * r38382922;
        double r38382924 = r38382915 + r38382900;
        double r38382925 = r38382923 / r38382924;
        double r38382926 = -r38382904;
        double r38382927 = r38382921 ? r38382925 : r38382926;
        double r38382928 = r38382909 ? r38382919 : r38382927;
        double r38382929 = r38382902 ? r38382907 : r38382928;
        return r38382929;
}

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

Original33.3
Target20.3
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.263941314600607e+152

    1. Initial program 60.1

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

    2. Simplified60.1

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

    3. Taylor expanded around -inf 2.3

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

    if -3.263941314600607e+152 < b < -4.687918346756617e-254

    1. Initial program 7.8

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

    2. Simplified7.8

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

    4. Applied clear-num8.0

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

    6. Applied *-un-lft-identity8.0

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

    7. Applied *-un-lft-identity8.0

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

    8. Applied distribute-lft-out--8.0

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

    9. Applied times-frac8.0

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

    10. Applied add-sqr-sqrt8.0

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

    11. Applied times-frac8.0

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

    12. Simplified8.0

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

    13. Simplified7.8

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

    if -4.687918346756617e-254 < b < 3.463606471108268e+121

    1. Initial program 31.6

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

    2. Simplified31.6

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

    4. Applied clear-num31.7

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

    6. Applied flip--31.8

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

    7. Applied associate-/r/31.9

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

    8. Applied associate-/r*31.9

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

    9. Simplified14.4

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

    10. Taylor expanded around inf 8.7

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

    if 3.463606471108268e+121 < b

    1. Initial program 59.8

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

    2. Simplified59.8

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

    3. Taylor expanded around inf 2.3

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

    4. Simplified2.3

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a} \cdot \frac{1}{2}\\ \mathbf{elif}\;b \le 3.463606471108268 \cdot 10^{+121}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))