Average Error: 33.6 → 10.4
Time: 20.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 0.17389787404847717:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -1.264659490877098 \cdot 10^{-67}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 0.17389787404847717:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3043923 = b;
        double r3043924 = -r3043923;
        double r3043925 = r3043923 * r3043923;
        double r3043926 = 4.0;
        double r3043927 = a;
        double r3043928 = c;
        double r3043929 = r3043927 * r3043928;
        double r3043930 = r3043926 * r3043929;
        double r3043931 = r3043925 - r3043930;
        double r3043932 = sqrt(r3043931);
        double r3043933 = r3043924 - r3043932;
        double r3043934 = 2.0;
        double r3043935 = r3043934 * r3043927;
        double r3043936 = r3043933 / r3043935;
        return r3043936;
}
↓
double f(double a, double b, double c) {
        double r3043937 = b;
        double r3043938 = -1.264659490877098e-67;
        bool r3043939 = r3043937 <= r3043938;
        double r3043940 = -2.0;
        double r3043941 = c;
        double r3043942 = r3043941 / r3043937;
        double r3043943 = r3043940 * r3043942;
        double r3043944 = 2.0;
        double r3043945 = r3043943 / r3043944;
        double r3043946 = 0.17389787404847717;
        bool r3043947 = r3043937 <= r3043946;
        double r3043948 = 1.0;
        double r3043949 = a;
        double r3043950 = r3043948 / r3043949;
        double r3043951 = -r3043937;
        double r3043952 = -4.0;
        double r3043953 = r3043949 * r3043952;
        double r3043954 = r3043937 * r3043937;
        double r3043955 = fma(r3043953, r3043941, r3043954);
        double r3043956 = sqrt(r3043955);
        double r3043957 = r3043951 - r3043956;
        double r3043958 = r3043950 * r3043957;
        double r3043959 = r3043958 / r3043944;
        double r3043960 = r3043937 / r3043949;
        double r3043961 = r3043942 - r3043960;
        double r3043962 = r3043961 * r3043944;
        double r3043963 = r3043962 / r3043944;
        double r3043964 = r3043947 ? r3043959 : r3043963;
        double r3043965 = r3043939 ? r3043945 : r3043964;
        return r3043965;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.9
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 < -1.264659490877098e-67

    1. Initial program 52.9

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

    2. Simplified53.0

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

    3. Taylor expanded around -inf 8.1

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

    if -1.264659490877098e-67 < b < 0.17389787404847717

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied div-inv15.1

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

    if 0.17389787404847717 < b

    1. Initial program 29.8

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

    2. Simplified29.8

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

    3. Taylor expanded around inf 7.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified7.3

      \[\leadsto \frac{\color{blue}{2 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 0.17389787404847717:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))