Average Error: 33.3 → 9.7
Time: 14.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.7874989996849275 \cdot 10^{-40}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.7665622931893247 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)}}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -5.7874989996849275 \cdot 10^{-40}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.7665622931893247 \cdot 10^{+83}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)}}{a} \cdot \frac{1}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1611765 = b;
        double r1611766 = -r1611765;
        double r1611767 = r1611765 * r1611765;
        double r1611768 = 4.0;
        double r1611769 = a;
        double r1611770 = c;
        double r1611771 = r1611769 * r1611770;
        double r1611772 = r1611768 * r1611771;
        double r1611773 = r1611767 - r1611772;
        double r1611774 = sqrt(r1611773);
        double r1611775 = r1611766 - r1611774;
        double r1611776 = 2.0;
        double r1611777 = r1611776 * r1611769;
        double r1611778 = r1611775 / r1611777;
        return r1611778;
}
↓
double f(double a, double b, double c) {
        double r1611779 = b;
        double r1611780 = -5.7874989996849275e-40;
        bool r1611781 = r1611779 <= r1611780;
        double r1611782 = c;
        double r1611783 = r1611782 / r1611779;
        double r1611784 = -r1611783;
        double r1611785 = 1.7665622931893247e+83;
        bool r1611786 = r1611779 <= r1611785;
        double r1611787 = -r1611779;
        double r1611788 = r1611779 * r1611779;
        double r1611789 = -4.0;
        double r1611790 = a;
        double r1611791 = r1611789 * r1611790;
        double r1611792 = r1611782 * r1611791;
        double r1611793 = r1611788 + r1611792;
        double r1611794 = sqrt(r1611793);
        double r1611795 = r1611787 - r1611794;
        double r1611796 = r1611795 / r1611790;
        double r1611797 = 0.5;
        double r1611798 = r1611796 * r1611797;
        double r1611799 = r1611779 / r1611790;
        double r1611800 = r1611783 - r1611799;
        double r1611801 = r1611786 ? r1611798 : r1611800;
        double r1611802 = r1611781 ? r1611784 : r1611801;
        return r1611802;
}

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

Original33.3
Target20.4
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.7874989996849275e-40

    1. Initial program 53.7

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

    2. Taylor expanded around -inf 7.3

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

    3. Simplified7.3

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

    if -5.7874989996849275e-40 < b < 1.7665622931893247e+83

    1. Initial program 13.8

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

    3. Applied clear-num13.9

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

    5. Applied *-un-lft-identity13.9

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

    6. Applied times-frac13.9

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

    7. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2}{1} \cdot \frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    8. Applied times-frac13.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    9. Simplified13.9

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    10. Simplified13.8

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

    if 1.7665622931893247e+83 < b

    1. Initial program 42.5

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

    2. Taylor expanded around inf 3.9

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.7874989996849275 \cdot 10^{-40}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.7665622931893247 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)}}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))