Average Error: 33.8 → 9.5
Time: 16.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.517926393801403 \cdot 10^{+142}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2796994 = b;
        double r2796995 = -r2796994;
        double r2796996 = r2796994 * r2796994;
        double r2796997 = 4.0;
        double r2796998 = a;
        double r2796999 = c;
        double r2797000 = r2796998 * r2796999;
        double r2797001 = r2796997 * r2797000;
        double r2797002 = r2796996 - r2797001;
        double r2797003 = sqrt(r2797002);
        double r2797004 = r2796995 + r2797003;
        double r2797005 = 2.0;
        double r2797006 = r2797005 * r2796998;
        double r2797007 = r2797004 / r2797006;
        return r2797007;
}
↓
double f(double a, double b, double c) {
        double r2797008 = b;
        double r2797009 = -5.517926393801403e+142;
        bool r2797010 = r2797008 <= r2797009;
        double r2797011 = c;
        double r2797012 = r2797011 / r2797008;
        double r2797013 = a;
        double r2797014 = r2797008 / r2797013;
        double r2797015 = r2797012 - r2797014;
        double r2797016 = 1.3635892865650846e-93;
        bool r2797017 = r2797008 <= r2797016;
        double r2797018 = r2797008 * r2797008;
        double r2797019 = 4.0;
        double r2797020 = r2797013 * r2797019;
        double r2797021 = r2797020 * r2797011;
        double r2797022 = r2797018 - r2797021;
        double r2797023 = sqrt(r2797022);
        double r2797024 = r2797023 - r2797008;
        double r2797025 = 0.5;
        double r2797026 = r2797025 / r2797013;
        double r2797027 = r2797024 * r2797026;
        double r2797028 = -r2797012;
        double r2797029 = r2797017 ? r2797027 : r2797028;
        double r2797030 = r2797010 ? r2797015 : r2797029;
        return r2797030;
}

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

Original33.8
Target20.5
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.517926393801403e+142

    1. Initial program 56.5

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

    2. Simplified56.5

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

    3. Taylor expanded around -inf 2.7

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

    if -5.517926393801403e+142 < b < 1.3635892865650846e-93

    1. Initial program 11.5

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

    2. Simplified11.6

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

    4. Applied *-un-lft-identity11.6

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

    5. Applied div-inv11.6

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

    6. Applied times-frac11.7

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

    7. Simplified11.7

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

    8. Simplified11.7

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

    if 1.3635892865650846e-93 < b

    1. Initial program 52.4

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

    2. Simplified52.4

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified9.1

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "quadp (p42, positive)"

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