Average Error: 34.2 → 6.7
Time: 6.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\left(\frac{2}{4} \cdot \frac{1}{c}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\
\;\;\;\;\frac{1}{\left(\frac{2}{4} \cdot \frac{1}{c}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r146800 = b;
        double r146801 = -r146800;
        double r146802 = r146800 * r146800;
        double r146803 = 4.0;
        double r146804 = a;
        double r146805 = r146803 * r146804;
        double r146806 = c;
        double r146807 = r146805 * r146806;
        double r146808 = r146802 - r146807;
        double r146809 = sqrt(r146808);
        double r146810 = r146801 + r146809;
        double r146811 = 2.0;
        double r146812 = r146811 * r146804;
        double r146813 = r146810 / r146812;
        return r146813;
}
↓
double f(double a, double b, double c) {
        double r146814 = b;
        double r146815 = -4.739386840053889e+131;
        bool r146816 = r146814 <= r146815;
        double r146817 = 1.0;
        double r146818 = c;
        double r146819 = r146818 / r146814;
        double r146820 = a;
        double r146821 = r146814 / r146820;
        double r146822 = r146819 - r146821;
        double r146823 = r146817 * r146822;
        double r146824 = -2.1023086245622604e-293;
        bool r146825 = r146814 <= r146824;
        double r146826 = -r146814;
        double r146827 = r146814 * r146814;
        double r146828 = 4.0;
        double r146829 = r146828 * r146820;
        double r146830 = r146829 * r146818;
        double r146831 = r146827 - r146830;
        double r146832 = sqrt(r146831);
        double r146833 = r146826 + r146832;
        double r146834 = 1.0;
        double r146835 = 2.0;
        double r146836 = r146835 * r146820;
        double r146837 = r146834 / r146836;
        double r146838 = r146833 * r146837;
        double r146839 = 6.09240124692818e+90;
        bool r146840 = r146814 <= r146839;
        double r146841 = r146835 / r146828;
        double r146842 = r146834 / r146818;
        double r146843 = r146841 * r146842;
        double r146844 = r146826 - r146832;
        double r146845 = r146843 * r146844;
        double r146846 = r146834 / r146845;
        double r146847 = -1.0;
        double r146848 = r146847 * r146819;
        double r146849 = r146840 ? r146846 : r146848;
        double r146850 = r146825 ? r146838 : r146849;
        double r146851 = r146816 ? r146823 : r146850;
        return r146851;
}

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

Original34.2
Target21.2
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.739386840053889e+131

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 2.4

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

    3. Simplified2.4

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

    if -4.739386840053889e+131 < b < -2.1023086245622604e-293

    1. Initial program 9.2

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

    3. Applied div-inv9.4

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

    if -2.1023086245622604e-293 < b < 6.09240124692818e+90

    1. Initial program 31.3

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

    3. Applied flip-+31.3

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

    4. Simplified16.0

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

    6. Applied clear-num16.2

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

    7. Simplified15.6

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

    9. Applied times-frac15.6

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

    10. Simplified8.8

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

    if 6.09240124692818e+90 < b

    1. Initial program 59.2

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

    2. Taylor expanded around inf 3.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\left(\frac{2}{4} \cdot \frac{1}{c}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))