Average Error: 20.1 → 8.6
Time: 41.3s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.900769547116861223219498082835437225018 \cdot 10^{46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-2, b, \frac{a}{b} \cdot \left(c \cdot 2\right)\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, b \cdot -2\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.900769547116861223219498082835437225018 \cdot 10^{46}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-2, b, \frac{a}{b} \cdot \left(c \cdot 2\right)\right)}\\

\end{array}\\

\mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, b \cdot -2\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\

\end{array}
double f(double a, double b, double c) {
        double r1176406 = b;
        double r1176407 = 0.0;
        bool r1176408 = r1176406 >= r1176407;
        double r1176409 = -r1176406;
        double r1176410 = r1176406 * r1176406;
        double r1176411 = 4.0;
        double r1176412 = a;
        double r1176413 = r1176411 * r1176412;
        double r1176414 = c;
        double r1176415 = r1176413 * r1176414;
        double r1176416 = r1176410 - r1176415;
        double r1176417 = sqrt(r1176416);
        double r1176418 = r1176409 - r1176417;
        double r1176419 = 2.0;
        double r1176420 = r1176419 * r1176412;
        double r1176421 = r1176418 / r1176420;
        double r1176422 = r1176419 * r1176414;
        double r1176423 = r1176409 + r1176417;
        double r1176424 = r1176422 / r1176423;
        double r1176425 = r1176408 ? r1176421 : r1176424;
        return r1176425;
}
↓
double f(double a, double b, double c) {
        double r1176426 = b;
        double r1176427 = -2.900769547116861e+46;
        bool r1176428 = r1176426 <= r1176427;
        double r1176429 = 0.0;
        bool r1176430 = r1176426 >= r1176429;
        double r1176431 = -r1176426;
        double r1176432 = r1176426 * r1176426;
        double r1176433 = 4.0;
        double r1176434 = a;
        double r1176435 = r1176433 * r1176434;
        double r1176436 = c;
        double r1176437 = r1176435 * r1176436;
        double r1176438 = r1176432 - r1176437;
        double r1176439 = sqrt(r1176438);
        double r1176440 = r1176431 - r1176439;
        double r1176441 = 2.0;
        double r1176442 = r1176441 * r1176434;
        double r1176443 = r1176440 / r1176442;
        double r1176444 = r1176436 * r1176441;
        double r1176445 = -2.0;
        double r1176446 = r1176434 / r1176426;
        double r1176447 = r1176446 * r1176444;
        double r1176448 = fma(r1176445, r1176426, r1176447);
        double r1176449 = r1176444 / r1176448;
        double r1176450 = r1176430 ? r1176443 : r1176449;
        double r1176451 = 0.17389787404847717;
        bool r1176452 = r1176426 <= r1176451;
        double r1176453 = sqrt(r1176439);
        double r1176454 = r1176453 * r1176453;
        double r1176455 = r1176431 + r1176454;
        double r1176456 = r1176444 / r1176455;
        double r1176457 = r1176430 ? r1176443 : r1176456;
        double r1176458 = r1176434 * r1176436;
        double r1176459 = r1176458 / r1176426;
        double r1176460 = r1176426 * r1176445;
        double r1176461 = fma(r1176441, r1176459, r1176460);
        double r1176462 = r1176461 / r1176442;
        double r1176463 = r1176439 + r1176431;
        double r1176464 = r1176444 / r1176463;
        double r1176465 = r1176430 ? r1176462 : r1176464;
        double r1176466 = r1176452 ? r1176457 : r1176465;
        double r1176467 = r1176428 ? r1176450 : r1176466;
        return r1176467;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.900769547116861e+46

    1. Initial program 25.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt25.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    4. Applied sqrt-prod25.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    5. Taylor expanded around -inf 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

    6. Simplified3.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(-2, b, \frac{a}{b} \cdot \left(c \cdot 2\right)\right)}}\\ \end{array}\]

    if -2.900769547116861e+46 < b < 0.17389787404847717

    1. Initial program 10.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt10.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    4. Applied sqrt-prod10.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    if 0.17389787404847717 < b

    1. Initial program 31.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    2. Taylor expanded around inf 11.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Simplified11.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, b \cdot -2\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.900769547116861223219498082835437225018 \cdot 10^{46}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(-2, b, \frac{a}{b} \cdot \left(c \cdot 2\right)\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, b \cdot -2\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))