Average Error: 34.1 → 9.9
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.260729798395838 \cdot 10^{-43}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 1.67643144401154069 \cdot 10^{104}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]
\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 -8.260729798395838 \cdot 10^{-43}:\\
\;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \le 1.67643144401154069 \cdot 10^{104}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r102899 = b;
        double r102900 = -r102899;
        double r102901 = r102899 * r102899;
        double r102902 = 4.0;
        double r102903 = a;
        double r102904 = c;
        double r102905 = r102903 * r102904;
        double r102906 = r102902 * r102905;
        double r102907 = r102901 - r102906;
        double r102908 = sqrt(r102907);
        double r102909 = r102900 - r102908;
        double r102910 = 2.0;
        double r102911 = r102910 * r102903;
        double r102912 = r102909 / r102911;
        return r102912;
}
↓
double f(double a, double b, double c) {
        double r102913 = b;
        double r102914 = -8.260729798395838e-43;
        bool r102915 = r102913 <= r102914;
        double r102916 = 1.0;
        double r102917 = -1.0;
        double r102918 = c;
        double r102919 = r102918 / r102913;
        double r102920 = r102917 * r102919;
        double r102921 = r102916 * r102920;
        double r102922 = 1.6764314440115407e+104;
        bool r102923 = r102913 <= r102922;
        double r102924 = -r102913;
        double r102925 = r102913 * r102913;
        double r102926 = 4.0;
        double r102927 = a;
        double r102928 = r102927 * r102918;
        double r102929 = r102926 * r102928;
        double r102930 = r102925 - r102929;
        double r102931 = sqrt(r102930);
        double r102932 = r102924 - r102931;
        double r102933 = 2.0;
        double r102934 = r102933 * r102927;
        double r102935 = r102932 / r102934;
        double r102936 = r102916 * r102935;
        double r102937 = 1.0;
        double r102938 = r102913 / r102927;
        double r102939 = r102919 - r102938;
        double r102940 = r102937 * r102939;
        double r102941 = r102916 * r102940;
        double r102942 = r102923 ? r102936 : r102941;
        double r102943 = r102915 ? r102921 : r102942;
        return r102943;
}

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

Original34.1
Target21.1
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.260729798395838e-43

    1. Initial program 54.7

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

    3. Applied div-inv54.7

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

    5. Applied *-un-lft-identity54.7

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

    6. Applied associate-*l*54.7

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

    7. Simplified54.7

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

    8. Taylor expanded around -inf 7.3

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

    if -8.260729798395838e-43 < b < 1.6764314440115407e+104

    1. Initial program 14.0

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

    3. Applied div-inv14.1

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

    5. Applied *-un-lft-identity14.1

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

    6. Applied associate-*l*14.1

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

    7. Simplified14.0

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

    if 1.6764314440115407e+104 < b

    1. Initial program 47.9

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

    3. Applied div-inv48.0

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

    5. Applied *-un-lft-identity48.0

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

    6. Applied associate-*l*48.0

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

    7. Simplified47.9

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

    8. Taylor expanded around inf 3.3

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

    9. Simplified3.3

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.260729798395838 \cdot 10^{-43}:\\ \;\;\;\;1 \cdot \left(-1 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 1.67643144401154069 \cdot 10^{104}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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)))