Average Error: 34.6 → 6.3
Time: 20.4s
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 r72959 = b;
        double r72960 = -r72959;
        double r72961 = r72959 * r72959;
        double r72962 = 4.0;
        double r72963 = a;
        double r72964 = c;
        double r72965 = r72963 * r72964;
        double r72966 = r72962 * r72965;
        double r72967 = r72961 - r72966;
        double r72968 = sqrt(r72967);
        double r72969 = r72960 - r72968;
        double r72970 = 2.0;
        double r72971 = r72970 * r72963;
        double r72972 = r72969 / r72971;
        return r72972;
}
↓
double f(double a, double b, double c) {
        double r72973 = b;
        double r72974 = -5.263290697710818e+146;
        bool r72975 = r72973 <= r72974;
        double r72976 = -1.0;
        double r72977 = c;
        double r72978 = r72977 / r72973;
        double r72979 = r72976 * r72978;
        double r72980 = -2.182382645844659e-295;
        bool r72981 = r72973 <= r72980;
        double r72982 = 1.0;
        double r72983 = 2.0;
        double r72984 = a;
        double r72985 = r72983 * r72984;
        double r72986 = 4.0;
        double r72987 = r72986 * r72984;
        double r72988 = r72985 / r72987;
        double r72989 = r72973 * r72973;
        double r72990 = r72984 * r72977;
        double r72991 = r72986 * r72990;
        double r72992 = r72989 - r72991;
        double r72993 = sqrt(r72992);
        double r72994 = r72993 - r72973;
        double r72995 = r72977 / r72994;
        double r72996 = r72988 / r72995;
        double r72997 = r72982 / r72996;
        double r72998 = 3.1607591925776442e+143;
        bool r72999 = r72973 <= r72998;
        double r73000 = -r72973;
        double r73001 = r73000 - r72993;
        double r73002 = r73001 / r72985;
        double r73003 = 1.0;
        double r73004 = r72973 / r72984;
        double r73005 = r72978 - r73004;
        double r73006 = r73003 * r73005;
        double r73007 = r72999 ? r73002 : r73006;
        double r73008 = r72981 ? r72997 : r73007;
        double r73009 = r72975 ? r72979 : r73008;
        return r73009;
}

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 clear-num15.9

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

    8. Simplified15.9

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

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

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

    11. Applied times-frac13.5

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

    12. Applied associate-/r*7.6

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

    13. Simplified7.6

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

    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 "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)))