Average Error: 33.2 → 10.7
Time: 19.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;-\frac{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} + b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;-\frac{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} + b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2230255 = b;
        double r2230256 = -r2230255;
        double r2230257 = r2230255 * r2230255;
        double r2230258 = 4.0;
        double r2230259 = a;
        double r2230260 = c;
        double r2230261 = r2230259 * r2230260;
        double r2230262 = r2230258 * r2230261;
        double r2230263 = r2230257 - r2230262;
        double r2230264 = sqrt(r2230263);
        double r2230265 = r2230256 - r2230264;
        double r2230266 = 2.0;
        double r2230267 = r2230266 * r2230259;
        double r2230268 = r2230265 / r2230267;
        return r2230268;
}
↓
double f(double a, double b, double c) {
        double r2230269 = b;
        double r2230270 = -2.2415082771065304e-131;
        bool r2230271 = r2230269 <= r2230270;
        double r2230272 = -2.0;
        double r2230273 = c;
        double r2230274 = r2230273 / r2230269;
        double r2230275 = r2230272 * r2230274;
        double r2230276 = 2.0;
        double r2230277 = r2230275 / r2230276;
        double r2230278 = 2.559678284282607e+69;
        bool r2230279 = r2230269 <= r2230278;
        double r2230280 = -4.0;
        double r2230281 = a;
        double r2230282 = r2230280 * r2230281;
        double r2230283 = r2230269 * r2230269;
        double r2230284 = fma(r2230273, r2230282, r2230283);
        double r2230285 = sqrt(r2230284);
        double r2230286 = r2230285 + r2230269;
        double r2230287 = r2230286 / r2230281;
        double r2230288 = r2230287 / r2230276;
        double r2230289 = -r2230288;
        double r2230290 = r2230269 / r2230281;
        double r2230291 = r2230274 - r2230290;
        double r2230292 = r2230291 * r2230276;
        double r2230293 = r2230292 / r2230276;
        double r2230294 = r2230279 ? r2230289 : r2230293;
        double r2230295 = r2230271 ? r2230277 : r2230294;
        return r2230295;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target19.9
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.2415082771065304e-131

    1. Initial program 49.6

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

    2. Simplified49.7

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

    3. Taylor expanded around -inf 12.4

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

    if -2.2415082771065304e-131 < b < 2.559678284282607e+69

    1. Initial program 11.4

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

    2. Simplified11.4

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

    4. Applied div-inv11.5

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

    6. Applied associate-*r/11.4

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

    7. Simplified11.4

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

    if 2.559678284282607e+69 < b

    1. Initial program 38.9

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

    2. Simplified38.9

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

    3. Taylor expanded around inf 4.8

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

    4. Simplified4.8

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;-\frac{\frac{\sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)} + b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))