Average Error: 34.5 → 6.7
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -6.31275544544182218 \cdot 10^{-270}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 6.7411875700484855 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.52779163831840318 \cdot 10^{117}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -6.31275544544182218 \cdot 10^{-270}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 6.7411875700484855 \cdot 10^{112}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r174902 = b;
        double r174903 = -r174902;
        double r174904 = r174902 * r174902;
        double r174905 = 4.0;
        double r174906 = a;
        double r174907 = r174905 * r174906;
        double r174908 = c;
        double r174909 = r174907 * r174908;
        double r174910 = r174904 - r174909;
        double r174911 = sqrt(r174910);
        double r174912 = r174903 + r174911;
        double r174913 = 2.0;
        double r174914 = r174913 * r174906;
        double r174915 = r174912 / r174914;
        return r174915;
}
↓
double f(double a, double b, double c) {
        double r174916 = b;
        double r174917 = -1.5277916383184032e+117;
        bool r174918 = r174916 <= r174917;
        double r174919 = 1.0;
        double r174920 = c;
        double r174921 = r174920 / r174916;
        double r174922 = a;
        double r174923 = r174916 / r174922;
        double r174924 = r174921 - r174923;
        double r174925 = r174919 * r174924;
        double r174926 = -6.312755445441822e-270;
        bool r174927 = r174916 <= r174926;
        double r174928 = 1.0;
        double r174929 = 2.0;
        double r174930 = r174929 * r174922;
        double r174931 = -r174916;
        double r174932 = r174916 * r174916;
        double r174933 = 4.0;
        double r174934 = r174933 * r174922;
        double r174935 = r174934 * r174920;
        double r174936 = r174932 - r174935;
        double r174937 = sqrt(r174936);
        double r174938 = r174931 + r174937;
        double r174939 = r174930 / r174938;
        double r174940 = r174928 / r174939;
        double r174941 = 6.7411875700484855e+112;
        bool r174942 = r174916 <= r174941;
        double r174943 = r174928 / r174929;
        double r174944 = r174920 * r174933;
        double r174945 = r174944 / r174928;
        double r174946 = r174931 - r174937;
        double r174947 = r174945 / r174946;
        double r174948 = r174943 * r174947;
        double r174949 = -1.0;
        double r174950 = r174949 * r174921;
        double r174951 = r174942 ? r174948 : r174950;
        double r174952 = r174927 ? r174940 : r174951;
        double r174953 = r174918 ? r174925 : r174952;
        return r174953;
}

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

Original34.5
Target21.4
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.5277916383184032e+117

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 3.7

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

    3. Simplified3.7

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

    if -1.5277916383184032e+117 < b < -6.312755445441822e-270

    1. Initial program 8.4

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

    3. Applied clear-num8.6

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

    if -6.312755445441822e-270 < b < 6.7411875700484855e+112

    1. Initial program 31.9

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

    3. Applied flip-+31.9

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

    4. Simplified16.8

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

    6. Applied *-un-lft-identity16.8

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

    7. Applied *-un-lft-identity16.8

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

    8. Applied times-frac16.8

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

    9. Applied times-frac16.8

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

    10. Simplified16.8

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

    11. Simplified21.4

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

    13. Applied associate-/r*16.0

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

    14. Simplified9.5

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

    if 6.7411875700484855e+112 < b

    1. Initial program 60.3

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

    2. Taylor expanded around inf 1.8

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -6.31275544544182218 \cdot 10^{-270}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 6.7411875700484855 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\frac{c \cdot 4}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))