Average Error: 33.2 → 9.7
Time: 27.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -4.170773079316174 \cdot 10^{+99}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3283078 = b;
        double r3283079 = -r3283078;
        double r3283080 = r3283078 * r3283078;
        double r3283081 = 4.0;
        double r3283082 = a;
        double r3283083 = r3283081 * r3283082;
        double r3283084 = c;
        double r3283085 = r3283083 * r3283084;
        double r3283086 = r3283080 - r3283085;
        double r3283087 = sqrt(r3283086);
        double r3283088 = r3283079 + r3283087;
        double r3283089 = 2.0;
        double r3283090 = r3283089 * r3283082;
        double r3283091 = r3283088 / r3283090;
        return r3283091;
}
↓
double f(double a, double b, double c) {
        double r3283092 = b;
        double r3283093 = -4.170773079316174e+99;
        bool r3283094 = r3283092 <= r3283093;
        double r3283095 = c;
        double r3283096 = r3283095 / r3283092;
        double r3283097 = a;
        double r3283098 = r3283092 / r3283097;
        double r3283099 = r3283096 - r3283098;
        double r3283100 = 2.0;
        double r3283101 = r3283099 * r3283100;
        double r3283102 = r3283101 / r3283100;
        double r3283103 = 3.0168583404714427e-66;
        bool r3283104 = r3283092 <= r3283103;
        double r3283105 = r3283095 * r3283097;
        double r3283106 = -4.0;
        double r3283107 = r3283105 * r3283106;
        double r3283108 = fma(r3283092, r3283092, r3283107);
        double r3283109 = sqrt(r3283108);
        double r3283110 = r3283109 / r3283097;
        double r3283111 = r3283110 - r3283098;
        double r3283112 = r3283111 / r3283100;
        double r3283113 = -2.0;
        double r3283114 = r3283113 * r3283096;
        double r3283115 = r3283114 / r3283100;
        double r3283116 = r3283104 ? r3283112 : r3283115;
        double r3283117 = r3283094 ? r3283102 : r3283116;
        return r3283117;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.2
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.170773079316174e+99

    1. Initial program 44.2

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

    2. Simplified44.2

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

    3. Taylor expanded around inf 44.2

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

    4. Taylor expanded around -inf 3.3

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

    5. Simplified3.3

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

    if -4.170773079316174e+99 < b < 3.0168583404714427e-66

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around inf 12.8

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

    5. Applied div-sub12.8

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

    if 3.0168583404714427e-66 < b

    1. Initial program 53.1

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

    2. Simplified53.1

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

    3. Taylor expanded around inf 53.0

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

    4. Taylor expanded around inf 8.5

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +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)))