Average Error: 34.4 → 9.1
Time: 5.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.47093887587755467 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{elif}\;b \le 1026034526.7184467:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.0461303908572575 \cdot 10^{65}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.47093887587755467 \cdot 10^{-146}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

\mathbf{elif}\;b \le 1026034526.7184467:\\
\;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r83074 = b;
        double r83075 = -r83074;
        double r83076 = r83074 * r83074;
        double r83077 = 4.0;
        double r83078 = a;
        double r83079 = c;
        double r83080 = r83078 * r83079;
        double r83081 = r83077 * r83080;
        double r83082 = r83076 - r83081;
        double r83083 = sqrt(r83082);
        double r83084 = r83075 + r83083;
        double r83085 = 2.0;
        double r83086 = r83085 * r83078;
        double r83087 = r83084 / r83086;
        return r83087;
}
↓
double f(double a, double b, double c) {
        double r83088 = b;
        double r83089 = -1.0461303908572575e+65;
        bool r83090 = r83088 <= r83089;
        double r83091 = 1.0;
        double r83092 = c;
        double r83093 = r83092 / r83088;
        double r83094 = a;
        double r83095 = r83088 / r83094;
        double r83096 = r83093 - r83095;
        double r83097 = r83091 * r83096;
        double r83098 = 3.4709388758775547e-146;
        bool r83099 = r83088 <= r83098;
        double r83100 = 1.0;
        double r83101 = 2.0;
        double r83102 = r83101 * r83094;
        double r83103 = -r83088;
        double r83104 = r83088 * r83088;
        double r83105 = 4.0;
        double r83106 = r83094 * r83092;
        double r83107 = r83105 * r83106;
        double r83108 = r83104 - r83107;
        double r83109 = sqrt(r83108);
        double r83110 = r83103 + r83109;
        double r83111 = r83102 / r83110;
        double r83112 = r83100 / r83111;
        double r83113 = 1026034526.7184467;
        bool r83114 = r83088 <= r83113;
        double r83115 = 0.0;
        double r83116 = r83115 + r83107;
        double r83117 = r83103 - r83109;
        double r83118 = r83116 / r83117;
        double r83119 = r83118 / r83102;
        double r83120 = -1.0;
        double r83121 = r83120 * r83093;
        double r83122 = r83114 ? r83119 : r83121;
        double r83123 = r83099 ? r83112 : r83122;
        double r83124 = r83090 ? r83097 : r83123;
        return r83124;
}

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

Original34.4
Target21.0
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.0461303908572575e+65

    1. Initial program 41.3

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

    2. Taylor expanded around -inf 4.6

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

    3. Simplified4.6

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

    if -1.0461303908572575e+65 < b < 3.4709388758775547e-146

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    if 3.4709388758775547e-146 < b < 1026034526.7184467

    1. Initial program 33.7

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

    3. Applied flip-+33.7

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

    4. Simplified17.5

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

    if 1026034526.7184467 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.47093887587755467 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{elif}\;b \le 1026034526.7184467:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))