Average Error: 19.2 → 6.5
Time: 31.4s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.148407540792454 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.6529445176198465 \cdot 10^{+100}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}, -b\right)}{2}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\ \end{array}\]
\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 -5.148407540792454 \cdot 10^{+110}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}{a \cdot 2}\\

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

\end{array}\\

\mathbf{elif}\;b \le 1.6529445176198465 \cdot 10^{+100}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{a \cdot 2}\\

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

\end{array}\\

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

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

\end{array}
double f(double a, double b, double c) {
        double r922279 = b;
        double r922280 = 0.0;
        bool r922281 = r922279 >= r922280;
        double r922282 = -r922279;
        double r922283 = r922279 * r922279;
        double r922284 = 4.0;
        double r922285 = a;
        double r922286 = r922284 * r922285;
        double r922287 = c;
        double r922288 = r922286 * r922287;
        double r922289 = r922283 - r922288;
        double r922290 = sqrt(r922289);
        double r922291 = r922282 - r922290;
        double r922292 = 2.0;
        double r922293 = r922292 * r922285;
        double r922294 = r922291 / r922293;
        double r922295 = r922292 * r922287;
        double r922296 = r922282 + r922290;
        double r922297 = r922295 / r922296;
        double r922298 = r922281 ? r922294 : r922297;
        return r922298;
}
↓
double f(double a, double b, double c) {
        double r922299 = b;
        double r922300 = -5.148407540792454e+110;
        bool r922301 = r922299 <= r922300;
        double r922302 = 0.0;
        bool r922303 = r922299 >= r922302;
        double r922304 = c;
        double r922305 = a;
        double r922306 = r922305 / r922299;
        double r922307 = r922304 * r922306;
        double r922308 = r922307 - r922299;
        double r922309 = 2.0;
        double r922310 = r922308 * r922309;
        double r922311 = r922305 * r922309;
        double r922312 = r922310 / r922311;
        double r922313 = r922304 * r922309;
        double r922314 = r922299 / r922305;
        double r922315 = r922304 / r922314;
        double r922316 = r922315 - r922299;
        double r922317 = r922309 * r922316;
        double r922318 = r922313 / r922317;
        double r922319 = r922303 ? r922312 : r922318;
        double r922320 = 1.6529445176198465e+100;
        bool r922321 = r922299 <= r922320;
        double r922322 = -r922299;
        double r922323 = -4.0;
        double r922324 = r922323 * r922305;
        double r922325 = r922299 * r922299;
        double r922326 = fma(r922304, r922324, r922325);
        double r922327 = sqrt(r922326);
        double r922328 = r922322 - r922327;
        double r922329 = r922328 / r922311;
        double r922330 = sqrt(r922327);
        double r922331 = fma(r922330, r922330, r922322);
        double r922332 = r922331 / r922309;
        double r922333 = r922304 / r922332;
        double r922334 = r922303 ? r922329 : r922333;
        double r922335 = r922321 ? r922334 : r922319;
        double r922336 = r922301 ? r922319 : r922335;
        return r922336;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < -5.148407540792454e+110 or 1.6529445176198465e+100 < b

    1. Initial program 36.6

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

    2. Taylor expanded around -inf 21.5

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

    3. Simplified19.3

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

    4. Taylor expanded around inf 5.2

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

    5. Simplified2.9

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

    if -5.148407540792454e+110 < b < 1.6529445176198465e+100

    1. Initial program 8.6

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

    2. Simplified8.6

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

    4. Applied add-sqr-sqrt8.6

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

    5. Applied sqrt-prod8.7

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

    6. Applied fma-neg8.7

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.148407540792454 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 1.6529445176198465 \cdot 10^{+100}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}, -b\right)}{2}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(c \cdot \frac{a}{b} - b\right) \cdot 2}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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)))))))