Average Error: 32.9 → 14.3
Time: 15.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 2.2058723073785985 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{a}}{2}\\ \end{array}\]
\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 -1.7512236628315378 \cdot 10^{+131}:\\
\;\;\;\;\frac{\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a}}{2}\\

\mathbf{elif}\;b \le 2.2058723073785985 \cdot 10^{-38}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2250465 = b;
        double r2250466 = -r2250465;
        double r2250467 = r2250465 * r2250465;
        double r2250468 = 4.0;
        double r2250469 = a;
        double r2250470 = r2250468 * r2250469;
        double r2250471 = c;
        double r2250472 = r2250470 * r2250471;
        double r2250473 = r2250467 - r2250472;
        double r2250474 = sqrt(r2250473);
        double r2250475 = r2250466 + r2250474;
        double r2250476 = 2.0;
        double r2250477 = r2250476 * r2250469;
        double r2250478 = r2250475 / r2250477;
        return r2250478;
}
↓
double f(double a, double b, double c) {
        double r2250479 = b;
        double r2250480 = -1.7512236628315378e+131;
        bool r2250481 = r2250479 <= r2250480;
        double r2250482 = a;
        double r2250483 = c;
        double r2250484 = r2250479 / r2250483;
        double r2250485 = r2250482 / r2250484;
        double r2250486 = r2250485 - r2250479;
        double r2250487 = 2.0;
        double r2250488 = r2250486 * r2250487;
        double r2250489 = r2250488 / r2250482;
        double r2250490 = r2250489 / r2250487;
        double r2250491 = 2.2058723073785985e-38;
        bool r2250492 = r2250479 <= r2250491;
        double r2250493 = -4.0;
        double r2250494 = r2250493 * r2250483;
        double r2250495 = r2250479 * r2250479;
        double r2250496 = fma(r2250482, r2250494, r2250495);
        double r2250497 = sqrt(r2250496);
        double r2250498 = r2250497 - r2250479;
        double r2250499 = r2250498 / r2250482;
        double r2250500 = r2250499 / r2250487;
        double r2250501 = -2.0;
        double r2250502 = r2250482 * r2250483;
        double r2250503 = r2250502 / r2250479;
        double r2250504 = r2250501 * r2250503;
        double r2250505 = r2250504 / r2250482;
        double r2250506 = r2250505 / r2250487;
        double r2250507 = r2250492 ? r2250500 : r2250506;
        double r2250508 = r2250481 ? r2250490 : r2250507;
        return r2250508;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.4
Herbie14.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7512236628315378e+131

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around -inf 9.8

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

    4. Simplified3.0

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

    if -1.7512236628315378e+131 < b < 2.2058723073785985e-38

    1. Initial program 13.7

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

    2. Simplified13.7

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

    3. Taylor expanded around 0 13.7

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

    4. Simplified13.7

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

    6. Applied *-un-lft-identity13.7

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

    if 2.2058723073785985e-38 < b

    1. Initial program 53.8

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

    2. Simplified53.8

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

    3. Taylor expanded around inf 19.0

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

  4. Final simplification14.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 2.2058723073785985 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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)))