Average Error: 34.5 → 10.5
Time: 8.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.78285893492843261 \cdot 10^{-126}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r73874 = b;
        double r73875 = -r73874;
        double r73876 = r73874 * r73874;
        double r73877 = 4.0;
        double r73878 = a;
        double r73879 = c;
        double r73880 = r73878 * r73879;
        double r73881 = r73877 * r73880;
        double r73882 = r73876 - r73881;
        double r73883 = sqrt(r73882);
        double r73884 = r73875 - r73883;
        double r73885 = 2.0;
        double r73886 = r73885 * r73878;
        double r73887 = r73884 / r73886;
        return r73887;
}
↓
double f(double a, double b, double c) {
        double r73888 = b;
        double r73889 = -4.7828589349284326e-126;
        bool r73890 = r73888 <= r73889;
        double r73891 = -1.0;
        double r73892 = c;
        double r73893 = r73892 / r73888;
        double r73894 = r73891 * r73893;
        double r73895 = 3.6627135292415903e+111;
        bool r73896 = r73888 <= r73895;
        double r73897 = -r73888;
        double r73898 = r73888 * r73888;
        double r73899 = 4.0;
        double r73900 = a;
        double r73901 = r73900 * r73892;
        double r73902 = r73899 * r73901;
        double r73903 = r73898 - r73902;
        double r73904 = sqrt(r73903);
        double r73905 = r73897 - r73904;
        double r73906 = 2.0;
        double r73907 = r73906 * r73900;
        double r73908 = r73905 / r73907;
        double r73909 = 1.0;
        double r73910 = r73888 / r73900;
        double r73911 = r73893 - r73910;
        double r73912 = r73909 * r73911;
        double r73913 = r73896 ? r73908 : r73912;
        double r73914 = r73890 ? r73894 : r73913;
        return r73914;
}

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

Original34.5
Target20.6
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.7828589349284326e-126

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 11.3

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

    if -4.7828589349284326e-126 < b < 3.6627135292415903e+111

    1. Initial program 12.0

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

    3. Applied div-inv12.1

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

    5. Applied *-un-lft-identity12.1

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

    6. Applied associate-*l*12.1

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

    7. Simplified12.0

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

    if 3.6627135292415903e+111 < b

    1. Initial program 49.7

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

    3. Applied div-inv49.8

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

    5. Applied *-un-lft-identity49.8

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

    6. Applied associate-*l*49.8

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

    7. Simplified49.7

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

    8. Taylor expanded around inf 3.4

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

    9. Simplified3.4

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.6627135292415903 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))