Average Error: 33.7 → 10.6
Time: 26.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3257822 = b;
        double r3257823 = -r3257822;
        double r3257824 = r3257822 * r3257822;
        double r3257825 = 4.0;
        double r3257826 = a;
        double r3257827 = c;
        double r3257828 = r3257826 * r3257827;
        double r3257829 = r3257825 * r3257828;
        double r3257830 = r3257824 - r3257829;
        double r3257831 = sqrt(r3257830);
        double r3257832 = r3257823 - r3257831;
        double r3257833 = 2.0;
        double r3257834 = r3257833 * r3257826;
        double r3257835 = r3257832 / r3257834;
        return r3257835;
}
↓
double f(double a, double b, double c) {
        double r3257836 = b;
        double r3257837 = -7.363255598823911e-15;
        bool r3257838 = r3257836 <= r3257837;
        double r3257839 = c;
        double r3257840 = -r3257839;
        double r3257841 = r3257840 / r3257836;
        double r3257842 = -6.936587154412951e-28;
        bool r3257843 = r3257836 <= r3257842;
        double r3257844 = -r3257836;
        double r3257845 = 2.0;
        double r3257846 = a;
        double r3257847 = r3257845 * r3257846;
        double r3257848 = r3257844 / r3257847;
        double r3257849 = 1.0;
        double r3257850 = r3257849 / r3257847;
        double r3257851 = r3257836 * r3257836;
        double r3257852 = r3257846 * r3257839;
        double r3257853 = 4.0;
        double r3257854 = r3257852 * r3257853;
        double r3257855 = r3257851 - r3257854;
        double r3257856 = sqrt(r3257855);
        double r3257857 = r3257850 * r3257856;
        double r3257858 = r3257848 - r3257857;
        double r3257859 = -2.3344326820285623e-123;
        bool r3257860 = r3257836 <= r3257859;
        double r3257861 = 1.6691257204922504e+85;
        bool r3257862 = r3257836 <= r3257861;
        double r3257863 = r3257847 / r3257856;
        double r3257864 = r3257849 / r3257863;
        double r3257865 = r3257848 - r3257864;
        double r3257866 = r3257839 / r3257836;
        double r3257867 = r3257836 / r3257846;
        double r3257868 = r3257866 - r3257867;
        double r3257869 = r3257862 ? r3257865 : r3257868;
        double r3257870 = r3257860 ? r3257841 : r3257869;
        double r3257871 = r3257843 ? r3257858 : r3257870;
        double r3257872 = r3257838 ? r3257841 : r3257871;
        return r3257872;
}

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
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    3. Applied div-sub51.4

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

    4. Taylor expanded around -inf 10.6

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

    5. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28

    1. Initial program 35.8

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

    3. Applied div-sub35.8

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

    5. Applied div-inv35.9

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

    if -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 12.6

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

    3. Applied div-sub12.6

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

    5. Applied clear-num12.7

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{2 \cdot a} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))