Average Error: 34.1 → 6.4
Time: 17.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.562666016748883260096099493207891090452 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.473620657135283571369379625694069201952 \cdot 10^{-291}:\\ \;\;\;\;\frac{1}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{\frac{4}{2} \cdot c}}\\ \mathbf{elif}\;b \le 5.810083498813659672832708321836264997329 \cdot 10^{102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -5.562666016748883260096099493207891090452 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.473620657135283571369379625694069201952 \cdot 10^{-291}:\\
\;\;\;\;\frac{1}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{\frac{4}{2} \cdot c}}\\

\mathbf{elif}\;b \le 5.810083498813659672832708321836264997329 \cdot 10^{102}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r61828 = b;
        double r61829 = -r61828;
        double r61830 = r61828 * r61828;
        double r61831 = 4.0;
        double r61832 = a;
        double r61833 = c;
        double r61834 = r61832 * r61833;
        double r61835 = r61831 * r61834;
        double r61836 = r61830 - r61835;
        double r61837 = sqrt(r61836);
        double r61838 = r61829 - r61837;
        double r61839 = 2.0;
        double r61840 = r61839 * r61832;
        double r61841 = r61838 / r61840;
        return r61841;
}
↓
double f(double a, double b, double c) {
        double r61842 = b;
        double r61843 = -5.562666016748883e+153;
        bool r61844 = r61842 <= r61843;
        double r61845 = -1.0;
        double r61846 = c;
        double r61847 = r61846 / r61842;
        double r61848 = r61845 * r61847;
        double r61849 = 1.4736206571352836e-291;
        bool r61850 = r61842 <= r61849;
        double r61851 = 1.0;
        double r61852 = r61842 * r61842;
        double r61853 = 4.0;
        double r61854 = a;
        double r61855 = r61854 * r61846;
        double r61856 = r61853 * r61855;
        double r61857 = r61852 - r61856;
        double r61858 = sqrt(r61857);
        double r61859 = r61858 - r61842;
        double r61860 = 2.0;
        double r61861 = r61853 / r61860;
        double r61862 = r61861 * r61846;
        double r61863 = r61851 / r61862;
        double r61864 = r61859 * r61863;
        double r61865 = r61851 / r61864;
        double r61866 = 5.81008349881366e+102;
        bool r61867 = r61842 <= r61866;
        double r61868 = -r61842;
        double r61869 = r61868 - r61858;
        double r61870 = r61860 * r61854;
        double r61871 = r61869 / r61870;
        double r61872 = -2.0;
        double r61873 = r61872 * r61842;
        double r61874 = r61873 / r61870;
        double r61875 = r61867 ? r61871 : r61874;
        double r61876 = r61850 ? r61865 : r61875;
        double r61877 = r61844 ? r61848 : r61876;
        return r61877;
}

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
Target20.7
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.562666016748883e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.5

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

    if -5.562666016748883e+153 < b < 1.4736206571352836e-291

    1. Initial program 34.3

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

    3. Applied flip--34.4

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

    4. Simplified15.7

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

    5. Simplified15.7

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

    7. Applied clear-num15.8

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

    8. Simplified15.8

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

    10. Applied associate-/r/14.5

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

    12. Applied clear-num14.5

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

    13. Simplified8.4

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

    if 1.4736206571352836e-291 < b < 5.81008349881366e+102

    1. Initial program 8.4

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

    if 5.81008349881366e+102 < b

    1. Initial program 47.7

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

    3. Applied flip--63.2

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    6. Taylor expanded around 0 3.4

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.562666016748883260096099493207891090452 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.473620657135283571369379625694069201952 \cdot 10^{-291}:\\ \;\;\;\;\frac{1}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{\frac{4}{2} \cdot c}}\\ \mathbf{elif}\;b \le 5.810083498813659672832708321836264997329 \cdot 10^{102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))