Average Error: 33.0 → 10.6
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.794505329565205 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.6194276288860963:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(a \cdot -4\right)\right)\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 r4042406 = b;
        double r4042407 = -r4042406;
        double r4042408 = r4042406 * r4042406;
        double r4042409 = 4.0;
        double r4042410 = a;
        double r4042411 = r4042409 * r4042410;
        double r4042412 = c;
        double r4042413 = r4042411 * r4042412;
        double r4042414 = r4042408 - r4042413;
        double r4042415 = sqrt(r4042414);
        double r4042416 = r4042407 + r4042415;
        double r4042417 = 2.0;
        double r4042418 = r4042417 * r4042410;
        double r4042419 = r4042416 / r4042418;
        return r4042419;
}
↓
double f(double a, double b, double c) {
        double r4042420 = b;
        double r4042421 = -3.794505329565205e+146;
        bool r4042422 = r4042420 <= r4042421;
        double r4042423 = c;
        double r4042424 = r4042423 / r4042420;
        double r4042425 = a;
        double r4042426 = r4042420 / r4042425;
        double r4042427 = r4042424 - r4042426;
        double r4042428 = 1.6194276288860963;
        bool r4042429 = r4042420 <= r4042428;
        double r4042430 = -4.0;
        double r4042431 = r4042425 * r4042430;
        double r4042432 = r4042423 * r4042431;
        double r4042433 = fma(r4042420, r4042420, r4042432);
        double r4042434 = sqrt(r4042433);
        double r4042435 = r4042434 - r4042420;
        double r4042436 = 2.0;
        double r4042437 = r4042435 / r4042436;
        double r4042438 = r4042437 / r4042425;
        double r4042439 = -r4042424;
        double r4042440 = r4042429 ? r4042438 : r4042439;
        double r4042441 = r4042422 ? r4042427 : r4042440;
        return r4042441;
}

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. Taylor expanded around 0 15.0

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

    4. Simplified15.0

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

    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 0 54.4

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

    4. Simplified54.4

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

    5. Taylor expanded around inf 5.9

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

    6. 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)))