Average Error: 34.6 → 8.7
Time: 18.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1150955755735961567232:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\
\;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r85332 = b;
        double r85333 = -r85332;
        double r85334 = r85332 * r85332;
        double r85335 = 4.0;
        double r85336 = a;
        double r85337 = c;
        double r85338 = r85336 * r85337;
        double r85339 = r85335 * r85338;
        double r85340 = r85334 - r85339;
        double r85341 = sqrt(r85340);
        double r85342 = r85333 - r85341;
        double r85343 = 2.0;
        double r85344 = r85343 * r85336;
        double r85345 = r85342 / r85344;
        return r85345;
}
↓
double f(double a, double b, double c) {
        double r85346 = b;
        double r85347 = -1.1509557557359616e+21;
        bool r85348 = r85346 <= r85347;
        double r85349 = -1.0;
        double r85350 = c;
        double r85351 = r85350 / r85346;
        double r85352 = r85349 * r85351;
        double r85353 = -3.1153949179978696e-213;
        bool r85354 = r85346 <= r85353;
        double r85355 = 4.0;
        double r85356 = a;
        double r85357 = r85356 * r85350;
        double r85358 = r85355 * r85357;
        double r85359 = -r85358;
        double r85360 = fma(r85346, r85346, r85359);
        double r85361 = sqrt(r85360);
        double r85362 = r85361 - r85346;
        double r85363 = r85358 / r85362;
        double r85364 = 1.0;
        double r85365 = 2.0;
        double r85366 = r85365 * r85356;
        double r85367 = r85364 / r85366;
        double r85368 = r85363 * r85367;
        double r85369 = 1.974261024048121e+145;
        bool r85370 = r85346 <= r85369;
        double r85371 = -r85346;
        double r85372 = r85346 * r85346;
        double r85373 = r85372 - r85358;
        double r85374 = sqrt(r85373);
        double r85375 = r85371 - r85374;
        double r85376 = r85375 / r85366;
        double r85377 = 1.0;
        double r85378 = r85346 / r85356;
        double r85379 = r85351 - r85378;
        double r85380 = r85377 * r85379;
        double r85381 = r85370 ? r85376 : r85380;
        double r85382 = r85354 ? r85368 : r85381;
        double r85383 = r85348 ? r85352 : r85382;
        return r85383;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.6
Target20.9
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1509557557359616e+21

    1. Initial program 56.3

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

    2. Taylor expanded around -inf 4.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -1.1509557557359616e+21 < b < -3.1153949179978696e-213

    1. Initial program 31.5

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

    3. Applied flip--31.5

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

    4. Simplified17.7

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified17.7

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv17.8

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]

    if -3.1153949179978696e-213 < b < 1.974261024048121e+145

    1. Initial program 9.9

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

    if 1.974261024048121e+145 < b

    1. Initial program 60.1

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

    2. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified2.3

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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