Average Error: 34.3 → 8.9
Time: 15.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.829903230896134050158793286773621805382 \cdot 10^{148}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.058331905530479345989188577279018272684 \cdot 10^{-144}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 400482480739649191422756162656796672:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.829903230896134050158793286773621805382 \cdot 10^{148}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.058331905530479345989188577279018272684 \cdot 10^{-144}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 400482480739649191422756162656796672:\\
\;\;\;\;\frac{\frac{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}\\

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

\end{array}
double f(double a, double b, double c) {
        double r42054 = b;
        double r42055 = -r42054;
        double r42056 = r42054 * r42054;
        double r42057 = 4.0;
        double r42058 = a;
        double r42059 = r42057 * r42058;
        double r42060 = c;
        double r42061 = r42059 * r42060;
        double r42062 = r42056 - r42061;
        double r42063 = sqrt(r42062);
        double r42064 = r42055 + r42063;
        double r42065 = 2.0;
        double r42066 = r42065 * r42058;
        double r42067 = r42064 / r42066;
        return r42067;
}
↓
double f(double a, double b, double c) {
        double r42068 = b;
        double r42069 = -4.829903230896134e+148;
        bool r42070 = r42068 <= r42069;
        double r42071 = 1.0;
        double r42072 = c;
        double r42073 = r42072 / r42068;
        double r42074 = a;
        double r42075 = r42068 / r42074;
        double r42076 = r42073 - r42075;
        double r42077 = r42071 * r42076;
        double r42078 = 1.0583319055304793e-144;
        bool r42079 = r42068 <= r42078;
        double r42080 = -r42068;
        double r42081 = r42068 * r42068;
        double r42082 = 4.0;
        double r42083 = r42082 * r42074;
        double r42084 = r42083 * r42072;
        double r42085 = r42081 - r42084;
        double r42086 = sqrt(r42085);
        double r42087 = r42080 + r42086;
        double r42088 = 2.0;
        double r42089 = r42088 * r42074;
        double r42090 = r42087 / r42089;
        double r42091 = 4.004824807396492e+35;
        bool r42092 = r42068 <= r42091;
        double r42093 = 0.0;
        double r42094 = r42074 * r42072;
        double r42095 = r42082 * r42094;
        double r42096 = r42093 + r42095;
        double r42097 = r42080 - r42086;
        double r42098 = r42096 / r42097;
        double r42099 = r42098 / r42089;
        double r42100 = -1.0;
        double r42101 = r42100 * r42073;
        double r42102 = r42092 ? r42099 : r42101;
        double r42103 = r42079 ? r42090 : r42102;
        double r42104 = r42070 ? r42077 : r42103;
        return r42104;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -4.829903230896134e+148

    1. Initial program 61.7

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -4.829903230896134e+148 < b < 1.0583319055304793e-144

    1. Initial program 11.2

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

    if 1.0583319055304793e-144 < b < 4.004824807396492e+35

    1. Initial program 35.7

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

    3. Applied flip-+35.8

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

    4. Simplified16.7

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

    if 4.004824807396492e+35 < b

    1. Initial program 56.6

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

    2. Taylor expanded around inf 4.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.829903230896134050158793286773621805382 \cdot 10^{148}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.058331905530479345989188577279018272684 \cdot 10^{-144}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 400482480739649191422756162656796672:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))