Average Error: 33.8 → 10.0
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r59318 = b;
        double r59319 = -r59318;
        double r59320 = r59318 * r59318;
        double r59321 = 4.0;
        double r59322 = a;
        double r59323 = r59321 * r59322;
        double r59324 = c;
        double r59325 = r59323 * r59324;
        double r59326 = r59320 - r59325;
        double r59327 = sqrt(r59326);
        double r59328 = r59319 + r59327;
        double r59329 = 2.0;
        double r59330 = r59329 * r59322;
        double r59331 = r59328 / r59330;
        return r59331;
}
↓
double f(double a, double b, double c) {
        double r59332 = b;
        double r59333 = -8.301687926884189e+98;
        bool r59334 = r59332 <= r59333;
        double r59335 = 1.0;
        double r59336 = 2.0;
        double r59337 = r59335 / r59336;
        double r59338 = c;
        double r59339 = r59338 / r59332;
        double r59340 = -2.0;
        double r59341 = a;
        double r59342 = r59332 / r59341;
        double r59343 = r59340 * r59342;
        double r59344 = fma(r59336, r59339, r59343);
        double r59345 = r59337 * r59344;
        double r59346 = 7.029337360841496e-56;
        bool r59347 = r59332 <= r59346;
        double r59348 = r59332 * r59332;
        double r59349 = 4.0;
        double r59350 = r59349 * r59341;
        double r59351 = r59350 * r59338;
        double r59352 = r59348 - r59351;
        double r59353 = sqrt(r59352);
        double r59354 = r59353 - r59332;
        double r59355 = r59354 / r59341;
        double r59356 = r59337 * r59355;
        double r59357 = -1.0;
        double r59358 = r59357 * r59339;
        double r59359 = r59347 ? r59356 : r59358;
        double r59360 = r59334 ? r59345 : r59359;
        return r59360;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.8
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.301687926884189e+98

    1. Initial program 46.2

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

    2. Simplified46.2

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

    4. Applied *-un-lft-identity46.2

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

    5. Applied times-frac46.2

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

    6. Taylor expanded around -inf 3.7

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

    7. Simplified3.7

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

    if -8.301687926884189e+98 < b < 7.029337360841496e-56

    1. Initial program 13.5

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

    2. Simplified13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied times-frac13.5

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

    if 7.029337360841496e-56 < 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

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

    3. Taylor expanded around inf 8.2

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.029337360841496098479843453825374035485 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))