Average Error: 33.1 → 8.5
Time: 50.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.9319920094724875 \cdot 10^{+55}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.516529709605321 \cdot 10^{-197}:\\ \;\;\;\;-\frac{\frac{\frac{a \cdot c}{\frac{-1}{2}}}{a}}{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}\\ \mathbf{elif}\;b \le 2.0734884367796945 \cdot 10^{+89}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 r11965804 = b;
        double r11965805 = -r11965804;
        double r11965806 = r11965804 * r11965804;
        double r11965807 = 4.0;
        double r11965808 = a;
        double r11965809 = c;
        double r11965810 = r11965808 * r11965809;
        double r11965811 = r11965807 * r11965810;
        double r11965812 = r11965806 - r11965811;
        double r11965813 = sqrt(r11965812);
        double r11965814 = r11965805 - r11965813;
        double r11965815 = 2.0;
        double r11965816 = r11965815 * r11965808;
        double r11965817 = r11965814 / r11965816;
        return r11965817;
}
↓
double f(double a, double b, double c) {
        double r11965818 = b;
        double r11965819 = -1.9319920094724875e+55;
        bool r11965820 = r11965818 <= r11965819;
        double r11965821 = c;
        double r11965822 = r11965821 / r11965818;
        double r11965823 = -r11965822;
        double r11965824 = -6.516529709605321e-197;
        bool r11965825 = r11965818 <= r11965824;
        double r11965826 = a;
        double r11965827 = r11965826 * r11965821;
        double r11965828 = -0.5;
        double r11965829 = r11965827 / r11965828;
        double r11965830 = r11965829 / r11965826;
        double r11965831 = -r11965818;
        double r11965832 = r11965818 * r11965818;
        double r11965833 = 4.0;
        double r11965834 = r11965827 * r11965833;
        double r11965835 = r11965832 - r11965834;
        double r11965836 = sqrt(r11965835);
        double r11965837 = r11965831 + r11965836;
        double r11965838 = r11965830 / r11965837;
        double r11965839 = -r11965838;
        double r11965840 = 2.0734884367796945e+89;
        bool r11965841 = r11965818 <= r11965840;
        double r11965842 = 2.0;
        double r11965843 = r11965826 * r11965842;
        double r11965844 = r11965831 / r11965843;
        double r11965845 = r11965836 / r11965843;
        double r11965846 = r11965844 - r11965845;
        double r11965847 = r11965818 / r11965826;
        double r11965848 = r11965822 - r11965847;
        double r11965849 = r11965841 ? r11965846 : r11965848;
        double r11965850 = r11965825 ? r11965839 : r11965849;
        double r11965851 = r11965820 ? r11965823 : r11965850;
        return r11965851;
}

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. Taylor expanded around -inf 3.7

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

    3. 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)))