Average Error: 34.6 → 6.3
Time: 19.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\
\;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\

\mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58912 = b;
        double r58913 = -r58912;
        double r58914 = r58912 * r58912;
        double r58915 = 4.0;
        double r58916 = a;
        double r58917 = c;
        double r58918 = r58916 * r58917;
        double r58919 = r58915 * r58918;
        double r58920 = r58914 - r58919;
        double r58921 = sqrt(r58920);
        double r58922 = r58913 - r58921;
        double r58923 = 2.0;
        double r58924 = r58923 * r58916;
        double r58925 = r58922 / r58924;
        return r58925;
}
↓
double f(double a, double b, double c) {
        double r58926 = b;
        double r58927 = -5.263290697710818e+146;
        bool r58928 = r58926 <= r58927;
        double r58929 = -1.0;
        double r58930 = c;
        double r58931 = r58930 / r58926;
        double r58932 = r58929 * r58931;
        double r58933 = -2.182382645844659e-295;
        bool r58934 = r58926 <= r58933;
        double r58935 = 1.0;
        double r58936 = 2.0;
        double r58937 = a;
        double r58938 = r58936 * r58937;
        double r58939 = 4.0;
        double r58940 = r58939 * r58937;
        double r58941 = r58938 / r58940;
        double r58942 = r58926 * r58926;
        double r58943 = r58937 * r58930;
        double r58944 = r58939 * r58943;
        double r58945 = r58942 - r58944;
        double r58946 = sqrt(r58945);
        double r58947 = r58946 - r58926;
        double r58948 = r58930 / r58947;
        double r58949 = r58941 / r58948;
        double r58950 = r58935 / r58949;
        double r58951 = 3.1607591925776442e+143;
        bool r58952 = r58926 <= r58951;
        double r58953 = -r58926;
        double r58954 = r58953 - r58946;
        double r58955 = r58954 / r58938;
        double r58956 = 1.0;
        double r58957 = r58926 / r58937;
        double r58958 = r58931 - r58957;
        double r58959 = r58956 * r58958;
        double r58960 = r58952 ? r58955 : r58959;
        double r58961 = r58934 ? r58950 : r58960;
        double r58962 = r58928 ? r58932 : r58961;
        return r58962;
}

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

Original34.6
Target20.9
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.263290697710818e+146

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 1.3

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

    if -5.263290697710818e+146 < b < -2.182382645844659e-295

    1. Initial program 34.7

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

    3. Applied flip--34.7

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

    4. Simplified15.7

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

    5. Simplified15.7

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

    7. Applied *-un-lft-identity15.7

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

    8. Applied *-un-lft-identity15.7

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

    9. Applied times-frac15.7

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

    10. Applied associate-/l*15.9

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

    11. Simplified15.9

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

    13. Applied *-un-lft-identity15.9

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

    14. Applied times-frac13.5

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

    15. Applied associate-/r*7.6

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

    16. Simplified7.6

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

    if -2.182382645844659e-295 < b < 3.1607591925776442e+143

    1. Initial program 9.3

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

    if 3.1607591925776442e+143 < b

    1. Initial program 59.6

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

    2. Taylor expanded around inf 2.3

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

    3. Simplified2.3

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))