Average Error: 34.1 → 10.6
Time: 13.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.151530580746178328361254057251815139505 \cdot 10^{-108}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 46522626219735482368:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -6.151530580746178328361254057251815139505 \cdot 10^{-108}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 46522626219735482368:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r104091 = b;
        double r104092 = -r104091;
        double r104093 = r104091 * r104091;
        double r104094 = 4.0;
        double r104095 = a;
        double r104096 = c;
        double r104097 = r104095 * r104096;
        double r104098 = r104094 * r104097;
        double r104099 = r104093 - r104098;
        double r104100 = sqrt(r104099);
        double r104101 = r104092 - r104100;
        double r104102 = 2.0;
        double r104103 = r104102 * r104095;
        double r104104 = r104101 / r104103;
        return r104104;
}
↓
double f(double a, double b, double c) {
        double r104105 = b;
        double r104106 = -6.151530580746178e-108;
        bool r104107 = r104105 <= r104106;
        double r104108 = -1.0;
        double r104109 = c;
        double r104110 = r104109 / r104105;
        double r104111 = r104108 * r104110;
        double r104112 = 4.652262621973548e+19;
        bool r104113 = r104105 <= r104112;
        double r104114 = 1.0;
        double r104115 = 2.0;
        double r104116 = a;
        double r104117 = r104115 * r104116;
        double r104118 = -r104105;
        double r104119 = r104105 * r104105;
        double r104120 = 4.0;
        double r104121 = r104116 * r104109;
        double r104122 = r104120 * r104121;
        double r104123 = r104119 - r104122;
        double r104124 = sqrt(r104123);
        double r104125 = r104118 - r104124;
        double r104126 = r104117 / r104125;
        double r104127 = r104114 / r104126;
        double r104128 = 1.0;
        double r104129 = r104105 / r104116;
        double r104130 = r104110 - r104129;
        double r104131 = r104128 * r104130;
        double r104132 = r104113 ? r104127 : r104131;
        double r104133 = r104107 ? r104111 : r104132;
        return r104133;
}

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

Original34.1
Target20.9
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.151530580746178e-108

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 10.7

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

    if -6.151530580746178e-108 < b < 4.652262621973548e+19

    1. Initial program 12.9

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

    3. Applied clear-num13.0

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

    if 4.652262621973548e+19 < b

    1. Initial program 34.5

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

    2. Taylor expanded around inf 6.9

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

    3. Simplified6.9

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.151530580746178328361254057251815139505 \cdot 10^{-108}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 46522626219735482368:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))