Average Error: 33.3 → 10.4
Time: 36.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -4.179137486378021 \cdot 10^{-24}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\
\;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2515887 = b;
        double r2515888 = -r2515887;
        double r2515889 = r2515887 * r2515887;
        double r2515890 = 4.0;
        double r2515891 = a;
        double r2515892 = c;
        double r2515893 = r2515891 * r2515892;
        double r2515894 = r2515890 * r2515893;
        double r2515895 = r2515889 - r2515894;
        double r2515896 = sqrt(r2515895);
        double r2515897 = r2515888 - r2515896;
        double r2515898 = 2.0;
        double r2515899 = r2515898 * r2515891;
        double r2515900 = r2515897 / r2515899;
        return r2515900;
}
↓
double f(double a, double b, double c) {
        double r2515901 = b;
        double r2515902 = -4.179137486378021e-24;
        bool r2515903 = r2515901 <= r2515902;
        double r2515904 = c;
        double r2515905 = r2515904 / r2515901;
        double r2515906 = -r2515905;
        double r2515907 = 2.3648644896474148e+52;
        bool r2515908 = r2515901 <= r2515907;
        double r2515909 = -r2515901;
        double r2515910 = a;
        double r2515911 = r2515904 * r2515910;
        double r2515912 = -4.0;
        double r2515913 = r2515901 * r2515901;
        double r2515914 = fma(r2515911, r2515912, r2515913);
        double r2515915 = sqrt(r2515914);
        double r2515916 = r2515909 - r2515915;
        double r2515917 = 0.5;
        double r2515918 = r2515916 * r2515917;
        double r2515919 = r2515918 / r2515910;
        double r2515920 = r2515901 / r2515910;
        double r2515921 = r2515905 - r2515920;
        double r2515922 = r2515908 ? r2515919 : r2515921;
        double r2515923 = r2515903 ? r2515906 : r2515922;
        return r2515923;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.7
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.179137486378021e-24

    1. Initial program 54.6

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

    3. Applied div-inv54.6

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

    4. Simplified54.6

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

    5. Taylor expanded around -inf 7.1

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

    6. Simplified7.1

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

    if -4.179137486378021e-24 < b < 2.3648644896474148e+52

    1. Initial program 15.1

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

    3. Applied div-inv15.3

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

    4. Simplified15.3

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

    6. Applied associate-*r/15.1

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

    7. Simplified15.1

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

    if 2.3648644896474148e+52 < b

    1. Initial program 36.7

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

    3. Applied div-inv36.9

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

    4. Simplified36.9

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

    6. Applied associate-*r/36.7

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

    7. Simplified36.7

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

    8. Taylor expanded around inf 5.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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