Average Error: 19.3 → 13.0
Time: 26.8s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3.514329182216526 \cdot 10^{+146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{e^{\log \left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right)}}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 3.514329182216526 \cdot 10^{+146}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}}{a \cdot 2}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot 2}{e^{\log \left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right)}}\\

\end{array}
double f(double a, double b, double c) {
        double r649341 = b;
        double r649342 = 0.0;
        bool r649343 = r649341 >= r649342;
        double r649344 = -r649341;
        double r649345 = r649341 * r649341;
        double r649346 = 4.0;
        double r649347 = a;
        double r649348 = r649346 * r649347;
        double r649349 = c;
        double r649350 = r649348 * r649349;
        double r649351 = r649345 - r649350;
        double r649352 = sqrt(r649351);
        double r649353 = r649344 - r649352;
        double r649354 = 2.0;
        double r649355 = r649354 * r649347;
        double r649356 = r649353 / r649355;
        double r649357 = r649354 * r649349;
        double r649358 = r649344 + r649352;
        double r649359 = r649357 / r649358;
        double r649360 = r649343 ? r649356 : r649359;
        return r649360;
}
↓
double f(double a, double b, double c) {
        double r649361 = b;
        double r649362 = 3.514329182216526e+146;
        bool r649363 = r649361 <= r649362;
        double r649364 = 0.0;
        bool r649365 = r649361 >= r649364;
        double r649366 = -r649361;
        double r649367 = a;
        double r649368 = -4.0;
        double r649369 = r649367 * r649368;
        double r649370 = c;
        double r649371 = r649361 * r649361;
        double r649372 = fma(r649369, r649370, r649371);
        double r649373 = sqrt(r649372);
        double r649374 = sqrt(r649373);
        double r649375 = r649374 * r649374;
        double r649376 = r649366 - r649375;
        double r649377 = 2.0;
        double r649378 = r649367 * r649377;
        double r649379 = r649376 / r649378;
        double r649380 = r649370 * r649377;
        double r649381 = r649373 - r649361;
        double r649382 = r649380 / r649381;
        double r649383 = r649365 ? r649379 : r649382;
        double r649384 = r649361 / r649370;
        double r649385 = r649367 / r649384;
        double r649386 = r649385 - r649361;
        double r649387 = r649386 * r649377;
        double r649388 = r649387 / r649378;
        double r649389 = log(r649381);
        double r649390 = exp(r649389);
        double r649391 = r649380 / r649390;
        double r649392 = r649365 ? r649388 : r649391;
        double r649393 = r649363 ? r649383 : r649392;
        return r649393;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 3.514329182216526e+146

    1. Initial program 14.3

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

    2. Simplified14.3

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt14.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\]

    5. Applied sqrt-prod14.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\]

    if 3.514329182216526e+146 < b

    1. Initial program 58.5

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

    2. Simplified58.5

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}}\]

    3. Taylor expanded around inf 10.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\]

    4. Simplified2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-exp-log2.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{e^{\log \left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b\right)}}}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.514329182216526 \cdot 10^{+146}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{e^{\log \left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))