Average Error: 33.0 → 10.6
Time: 24.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(a \cdot -4\right)\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.6194276288860963:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(a \cdot -4\right)\right)\right)} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2402928 = b;
        double r2402929 = -r2402928;
        double r2402930 = r2402928 * r2402928;
        double r2402931 = 4.0;
        double r2402932 = a;
        double r2402933 = r2402931 * r2402932;
        double r2402934 = c;
        double r2402935 = r2402933 * r2402934;
        double r2402936 = r2402930 - r2402935;
        double r2402937 = sqrt(r2402936);
        double r2402938 = r2402929 + r2402937;
        double r2402939 = 2.0;
        double r2402940 = r2402939 * r2402932;
        double r2402941 = r2402938 / r2402940;
        return r2402941;
}
↓
double f(double a, double b, double c) {
        double r2402942 = b;
        double r2402943 = -3.794505329565205e+146;
        bool r2402944 = r2402942 <= r2402943;
        double r2402945 = c;
        double r2402946 = r2402945 / r2402942;
        double r2402947 = a;
        double r2402948 = r2402942 / r2402947;
        double r2402949 = r2402946 - r2402948;
        double r2402950 = 1.6194276288860963;
        bool r2402951 = r2402942 <= r2402950;
        double r2402952 = -4.0;
        double r2402953 = r2402947 * r2402952;
        double r2402954 = r2402945 * r2402953;
        double r2402955 = fma(r2402942, r2402942, r2402954);
        double r2402956 = sqrt(r2402955);
        double r2402957 = r2402956 - r2402942;
        double r2402958 = 2.0;
        double r2402959 = r2402957 / r2402958;
        double r2402960 = r2402959 / r2402947;
        double r2402961 = -r2402946;
        double r2402962 = r2402951 ? r2402960 : r2402961;
        double r2402963 = r2402944 ? r2402949 : r2402962;
        return r2402963;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.0
Target20.2
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.794505329565205e+146

    1. Initial program 58.0

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

    2. Simplified58.0

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

    3. Taylor expanded around -inf 3.1

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

    if -3.794505329565205e+146 < b < 1.6194276288860963

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied *-un-lft-identity15.0

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

    5. Applied associate-/r*15.0

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

    6. Simplified15.0

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

    if 1.6194276288860963 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 5.9

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

    4. Simplified5.9

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(a \cdot -4\right)\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))