Average Error: 34.5 → 10.1
Time: 16.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 9.146553797812776905149011816443241056322 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 9.146553797812776905149011816443241056322 \cdot 10^{-50}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r60488 = b;
        double r60489 = -r60488;
        double r60490 = r60488 * r60488;
        double r60491 = 4.0;
        double r60492 = a;
        double r60493 = r60491 * r60492;
        double r60494 = c;
        double r60495 = r60493 * r60494;
        double r60496 = r60490 - r60495;
        double r60497 = sqrt(r60496);
        double r60498 = r60489 + r60497;
        double r60499 = 2.0;
        double r60500 = r60499 * r60492;
        double r60501 = r60498 / r60500;
        return r60501;
}
↓
double f(double a, double b, double c) {
        double r60502 = b;
        double r60503 = -1.8572382657132166e+109;
        bool r60504 = r60502 <= r60503;
        double r60505 = 1.0;
        double r60506 = c;
        double r60507 = r60506 / r60502;
        double r60508 = a;
        double r60509 = r60502 / r60508;
        double r60510 = r60507 - r60509;
        double r60511 = r60505 * r60510;
        double r60512 = 9.146553797812777e-50;
        bool r60513 = r60502 <= r60512;
        double r60514 = 1.0;
        double r60515 = 2.0;
        double r60516 = r60515 * r60508;
        double r60517 = r60502 * r60502;
        double r60518 = 4.0;
        double r60519 = r60518 * r60508;
        double r60520 = r60519 * r60506;
        double r60521 = r60517 - r60520;
        double r60522 = sqrt(r60521);
        double r60523 = r60522 - r60502;
        double r60524 = r60516 / r60523;
        double r60525 = r60514 / r60524;
        double r60526 = -1.0;
        double r60527 = r60526 * r60507;
        double r60528 = r60513 ? r60525 : r60527;
        double r60529 = r60504 ? r60511 : r60528;
        return r60529;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.5
Target20.8
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8572382657132166e+109

    1. Initial program 50.1

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

    2. Simplified50.1

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -1.8572382657132166e+109 < b < 9.146553797812777e-50

    1. Initial program 13.8

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

    2. Simplified13.8

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

    4. Applied clear-num13.9

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

    if 9.146553797812777e-50 < b

    1. Initial program 54.3

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

    2. Simplified54.3

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

    3. Taylor expanded around inf 7.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 9.146553797812776905149011816443241056322 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))