Average Error: 33.3 → 10.0
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -4.82289647433212 \cdot 10^{+153}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3966005 = b;
        double r3966006 = -r3966005;
        double r3966007 = r3966005 * r3966005;
        double r3966008 = 4.0;
        double r3966009 = a;
        double r3966010 = r3966008 * r3966009;
        double r3966011 = c;
        double r3966012 = r3966010 * r3966011;
        double r3966013 = r3966007 - r3966012;
        double r3966014 = sqrt(r3966013);
        double r3966015 = r3966006 + r3966014;
        double r3966016 = 2.0;
        double r3966017 = r3966016 * r3966009;
        double r3966018 = r3966015 / r3966017;
        return r3966018;
}
↓
double f(double a, double b, double c) {
        double r3966019 = b;
        double r3966020 = -4.82289647433212e+153;
        bool r3966021 = r3966019 <= r3966020;
        double r3966022 = c;
        double r3966023 = r3966022 / r3966019;
        double r3966024 = a;
        double r3966025 = r3966019 / r3966024;
        double r3966026 = r3966023 - r3966025;
        double r3966027 = 2.0;
        double r3966028 = r3966026 * r3966027;
        double r3966029 = r3966028 / r3966027;
        double r3966030 = 3.289226058156428e-70;
        bool r3966031 = r3966019 <= r3966030;
        double r3966032 = -4.0;
        double r3966033 = r3966032 * r3966024;
        double r3966034 = r3966033 * r3966022;
        double r3966035 = fma(r3966019, r3966019, r3966034);
        double r3966036 = sqrt(r3966035);
        double r3966037 = r3966036 - r3966019;
        double r3966038 = r3966037 / r3966024;
        double r3966039 = r3966038 / r3966027;
        double r3966040 = -2.0;
        double r3966041 = r3966023 * r3966040;
        double r3966042 = r3966041 / r3966027;
        double r3966043 = r3966031 ? r3966039 : r3966042;
        double r3966044 = r3966021 ? r3966029 : r3966043;
        return r3966044;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

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

Derivation

  1. Split input into 3 regimes

  2. if b < -4.82289647433212e+153

    1. Initial program 60.9

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

    2. Simplified60.9

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

    4. Applied div-inv60.9

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

    6. Applied div-inv60.9

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

    7. Applied associate-*r*60.9

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

    8. Simplified60.9

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

    9. Taylor expanded around -inf 2.3

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

    10. Simplified2.3

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

    if -4.82289647433212e+153 < b < 3.289226058156428e-70

    1. Initial program 12.4

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

    2. Simplified12.4

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

    4. Applied div-inv12.6

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

    6. Applied un-div-inv12.4

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

    if 3.289226058156428e-70 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    4. Applied div-inv52.2

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

    6. Applied div-inv52.2

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

    7. Applied associate-*r*52.2

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

    8. Simplified52.2

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

    9. Taylor expanded around inf 9.1

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +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)))