Average Error: 33.4 → 9.8
Time: 31.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.852138444177435 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{a \cdot \frac{2}{-\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.852138444177435 \cdot 10^{-54}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\
\;\;\;\;\frac{1}{a \cdot \frac{2}{-\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} + b\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r3877320 = b;
        double r3877321 = -r3877320;
        double r3877322 = r3877320 * r3877320;
        double r3877323 = 4.0;
        double r3877324 = a;
        double r3877325 = c;
        double r3877326 = r3877324 * r3877325;
        double r3877327 = r3877323 * r3877326;
        double r3877328 = r3877322 - r3877327;
        double r3877329 = sqrt(r3877328);
        double r3877330 = r3877321 - r3877329;
        double r3877331 = 2.0;
        double r3877332 = r3877331 * r3877324;
        double r3877333 = r3877330 / r3877332;
        return r3877333;
}
↓
double f(double a, double b, double c) {
        double r3877334 = b;
        double r3877335 = -2.852138444177435e-54;
        bool r3877336 = r3877334 <= r3877335;
        double r3877337 = c;
        double r3877338 = r3877337 / r3877334;
        double r3877339 = -r3877338;
        double r3877340 = 6.359263193477048e+137;
        bool r3877341 = r3877334 <= r3877340;
        double r3877342 = 1.0;
        double r3877343 = a;
        double r3877344 = 2.0;
        double r3877345 = r3877337 * r3877343;
        double r3877346 = -4.0;
        double r3877347 = r3877345 * r3877346;
        double r3877348 = fma(r3877334, r3877334, r3877347);
        double r3877349 = sqrt(r3877348);
        double r3877350 = r3877349 + r3877334;
        double r3877351 = -r3877350;
        double r3877352 = r3877344 / r3877351;
        double r3877353 = r3877343 * r3877352;
        double r3877354 = r3877342 / r3877353;
        double r3877355 = r3877334 / r3877343;
        double r3877356 = r3877338 - r3877355;
        double r3877357 = r3877341 ? r3877354 : r3877356;
        double r3877358 = r3877336 ? r3877339 : r3877357;
        return r3877358;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.8
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.852138444177435e-54

    1. Initial program 53.4

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

    3. Applied div-inv53.4

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

    4. Simplified53.4

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

    6. Applied fma-neg53.4

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

    7. Simplified53.4

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

    8. Taylor expanded around -inf 8.3

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

    9. Simplified8.3

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

    if -2.852138444177435e-54 < b < 6.359263193477048e+137

    1. Initial program 12.6

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

    3. Applied clear-num12.7

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

    4. Simplified12.8

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

    if 6.359263193477048e+137 < b

    1. Initial program 53.0

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.852138444177435 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{a \cdot \frac{2}{-\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +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)))