Average Error: 33.7 → 9.9
Time: 19.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.13576834014972722 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.45462261334491207 \cdot 10^{104}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.13576834014972722 \cdot 10^{-82}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 7.45462261334491207 \cdot 10^{104}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r53913 = b;
        double r53914 = -r53913;
        double r53915 = r53913 * r53913;
        double r53916 = 4.0;
        double r53917 = a;
        double r53918 = c;
        double r53919 = r53917 * r53918;
        double r53920 = r53916 * r53919;
        double r53921 = r53915 - r53920;
        double r53922 = sqrt(r53921);
        double r53923 = r53914 - r53922;
        double r53924 = 2.0;
        double r53925 = r53924 * r53917;
        double r53926 = r53923 / r53925;
        return r53926;
}
↓
double f(double a, double b, double c) {
        double r53927 = b;
        double r53928 = -1.1357683401497272e-82;
        bool r53929 = r53927 <= r53928;
        double r53930 = -1.0;
        double r53931 = c;
        double r53932 = r53931 / r53927;
        double r53933 = r53930 * r53932;
        double r53934 = 7.454622613344912e+104;
        bool r53935 = r53927 <= r53934;
        double r53936 = 1.0;
        double r53937 = 2.0;
        double r53938 = a;
        double r53939 = r53937 * r53938;
        double r53940 = -r53927;
        double r53941 = r53927 * r53927;
        double r53942 = 4.0;
        double r53943 = r53938 * r53931;
        double r53944 = r53942 * r53943;
        double r53945 = r53941 - r53944;
        double r53946 = sqrt(r53945);
        double r53947 = r53940 - r53946;
        double r53948 = r53939 / r53947;
        double r53949 = r53936 / r53948;
        double r53950 = 1.0;
        double r53951 = r53927 / r53938;
        double r53952 = r53932 - r53951;
        double r53953 = r53950 * r53952;
        double r53954 = r53935 ? r53949 : r53953;
        double r53955 = r53929 ? r53933 : r53954;
        return r53955;
}

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

Original33.7
Target20.6
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 < -1.1357683401497272e-82

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 9.6

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

    if -1.1357683401497272e-82 < b < 7.454622613344912e+104

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 7.454622613344912e+104 < b

    1. Initial program 49.0

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.13576834014972722 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.45462261334491207 \cdot 10^{104}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))