Average Error: 34.1 → 6.4
Time: 18.0s
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 r91769 = b;
        double r91770 = -r91769;
        double r91771 = r91769 * r91769;
        double r91772 = 4.0;
        double r91773 = a;
        double r91774 = c;
        double r91775 = r91773 * r91774;
        double r91776 = r91772 * r91775;
        double r91777 = r91771 - r91776;
        double r91778 = sqrt(r91777);
        double r91779 = r91770 - r91778;
        double r91780 = 2.0;
        double r91781 = r91780 * r91773;
        double r91782 = r91779 / r91781;
        return r91782;
}
↓
double f(double a, double b, double c) {
        double r91783 = b;
        double r91784 = -5.562666016748883e+153;
        bool r91785 = r91783 <= r91784;
        double r91786 = -1.0;
        double r91787 = c;
        double r91788 = r91787 / r91783;
        double r91789 = r91786 * r91788;
        double r91790 = 1.4736206571352836e-291;
        bool r91791 = r91783 <= r91790;
        double r91792 = 1.0;
        double r91793 = r91783 * r91783;
        double r91794 = 4.0;
        double r91795 = a;
        double r91796 = r91795 * r91787;
        double r91797 = r91794 * r91796;
        double r91798 = r91793 - r91797;
        double r91799 = sqrt(r91798);
        double r91800 = r91799 - r91783;
        double r91801 = 2.0;
        double r91802 = r91794 / r91801;
        double r91803 = r91802 * r91787;
        double r91804 = r91792 / r91803;
        double r91805 = r91800 * r91804;
        double r91806 = r91792 / r91805;
        double r91807 = 5.81008349881366e+102;
        bool r91808 = r91783 <= r91807;
        double r91809 = -r91783;
        double r91810 = r91809 - r91799;
        double r91811 = r91801 * r91795;
        double r91812 = r91810 / r91811;
        double r91813 = -2.0;
        double r91814 = r91813 * r91783;
        double r91815 = r91814 / r91811;
        double r91816 = r91808 ? r91812 : r91815;
        double r91817 = r91791 ? r91806 : r91816;
        double r91818 = r91785 ? r91789 : r91817;
        return r91818;
}

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 "The quadratic formula (r2)"
  :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)))