Average Error: 33.9 → 11.3
Time: 17.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58828 = b;
        double r58829 = -r58828;
        double r58830 = r58828 * r58828;
        double r58831 = 4.0;
        double r58832 = a;
        double r58833 = c;
        double r58834 = r58832 * r58833;
        double r58835 = r58831 * r58834;
        double r58836 = r58830 - r58835;
        double r58837 = sqrt(r58836);
        double r58838 = r58829 - r58837;
        double r58839 = 2.0;
        double r58840 = r58839 * r58832;
        double r58841 = r58838 / r58840;
        return r58841;
}
↓
double f(double a, double b, double c) {
        double r58842 = b;
        double r58843 = -1.8696623466311214e+101;
        bool r58844 = r58842 <= r58843;
        double r58845 = -1.0;
        double r58846 = c;
        double r58847 = r58846 / r58842;
        double r58848 = r58845 * r58847;
        double r58849 = 7.455592343308264e-170;
        bool r58850 = r58842 <= r58849;
        double r58851 = 2.0;
        double r58852 = r58851 * r58846;
        double r58853 = 4.0;
        double r58854 = a;
        double r58855 = r58854 * r58846;
        double r58856 = r58853 * r58855;
        double r58857 = -r58856;
        double r58858 = fma(r58842, r58842, r58857);
        double r58859 = sqrt(r58858);
        double r58860 = r58859 - r58842;
        double r58861 = r58852 / r58860;
        double r58862 = 1.0;
        double r58863 = r58842 / r58854;
        double r58864 = r58847 - r58863;
        double r58865 = r58862 * r58864;
        double r58866 = r58850 ? r58861 : r58865;
        double r58867 = r58844 ? r58848 : r58866;
        return r58867;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.9
Target21.1
Herbie11.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8696623466311214e+101

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 2.5

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

    if -1.8696623466311214e+101 < b < 7.455592343308264e-170

    1. Initial program 28.9

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

    3. Applied flip--29.1

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

    4. Simplified16.7

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

    5. Simplified16.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv16.7

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]
    8. Using strategy rm

    9. Applied associate-*l/16.2

      \[\leadsto \color{blue}{\frac{\left(0 + \left(a \cdot c\right) \cdot 4\right) \cdot \frac{1}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}\]

    10. Simplified16.1

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

    11. Taylor expanded around 0 11.1

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

    if 7.455592343308264e-170 < b

    1. Initial program 23.0

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

    2. Taylor expanded around inf 17.1

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

    3. Simplified17.1

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.869662346631121401645595393947635525169 \cdot 10^{101}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))