Average Error: 33.1 → 8.5
Time: 52.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.9319920094724875 \cdot 10^{+55}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\ \;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.9319920094724875 \cdot 10^{+55}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\
\;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\

\mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\
\;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r11965809 = b;
        double r11965810 = -r11965809;
        double r11965811 = r11965809 * r11965809;
        double r11965812 = 4.0;
        double r11965813 = a;
        double r11965814 = c;
        double r11965815 = r11965813 * r11965814;
        double r11965816 = r11965812 * r11965815;
        double r11965817 = r11965811 - r11965816;
        double r11965818 = sqrt(r11965817);
        double r11965819 = r11965810 - r11965818;
        double r11965820 = 2.0;
        double r11965821 = r11965820 * r11965813;
        double r11965822 = r11965819 / r11965821;
        return r11965822;
}
↓
double f(double a, double b, double c) {
        double r11965823 = b;
        double r11965824 = -1.9319920094724875e+55;
        bool r11965825 = r11965823 <= r11965824;
        double r11965826 = c;
        double r11965827 = r11965826 / r11965823;
        double r11965828 = -r11965827;
        double r11965829 = -6.516529709605321e-197;
        bool r11965830 = r11965823 <= r11965829;
        double r11965831 = a;
        double r11965832 = r11965831 * r11965826;
        double r11965833 = -0.5;
        double r11965834 = r11965832 / r11965833;
        double r11965835 = r11965834 / r11965831;
        double r11965836 = -r11965823;
        double r11965837 = r11965823 * r11965823;
        double r11965838 = 4.0;
        double r11965839 = r11965832 * r11965838;
        double r11965840 = r11965837 - r11965839;
        double r11965841 = sqrt(r11965840);
        double r11965842 = r11965836 + r11965841;
        double r11965843 = r11965835 / r11965842;
        double r11965844 = -r11965843;
        double r11965845 = 2.0734884367796945e+89;
        bool r11965846 = r11965823 <= r11965845;
        double r11965847 = 2.0;
        double r11965848 = r11965831 * r11965847;
        double r11965849 = r11965836 / r11965848;
        double r11965850 = r11965841 / r11965848;
        double r11965851 = r11965849 - r11965850;
        double r11965852 = r11965823 / r11965831;
        double r11965853 = r11965827 - r11965852;
        double r11965854 = r11965846 ? r11965851 : r11965853;
        double r11965855 = r11965830 ? r11965844 : r11965854;
        double r11965856 = r11965825 ? r11965828 : r11965855;
        return r11965856;
}

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

Original33.1
Target20.3
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.9319920094724875e+55

    1. Initial program 56.5

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

    3. Applied div-inv56.5

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

    4. Taylor expanded around -inf 3.7

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

    5. Simplified3.7

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

    if -1.9319920094724875e+55 < b < -6.516529709605321e-197

    1. Initial program 34.0

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

    3. Applied div-inv34.1

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

    5. Applied flip--34.2

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

    6. Applied associate-*l/34.2

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

    7. Simplified15.9

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

    if -6.516529709605321e-197 < b < 2.0734884367796945e+89

    1. Initial program 10.3

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

    3. Applied div-sub10.3

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

    if 2.0734884367796945e+89 < b

    1. Initial program 42.0

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

    3. Applied div-inv42.1

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

    4. Taylor expanded around inf 4.1

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.9319920094724875 \cdot 10^{+55}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\ \;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(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)))