Average Error: 34.3 → 6.6
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.941529513459981659526481896938835655376 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.752150488567700802544439975375947245276 \cdot 10^{103}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.297522851756149307625287590446857172218 \cdot 10^{130}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.941529513459981659526481896938835655376 \cdot 10^{-260}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.752150488567700802544439975375947245276 \cdot 10^{103}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r97486 = b;
        double r97487 = -r97486;
        double r97488 = r97486 * r97486;
        double r97489 = 4.0;
        double r97490 = a;
        double r97491 = r97489 * r97490;
        double r97492 = c;
        double r97493 = r97491 * r97492;
        double r97494 = r97488 - r97493;
        double r97495 = sqrt(r97494);
        double r97496 = r97487 + r97495;
        double r97497 = 2.0;
        double r97498 = r97497 * r97490;
        double r97499 = r97496 / r97498;
        return r97499;
}
↓
double f(double a, double b, double c) {
        double r97500 = b;
        double r97501 = -4.2975228517561493e+130;
        bool r97502 = r97500 <= r97501;
        double r97503 = 1.0;
        double r97504 = c;
        double r97505 = r97504 / r97500;
        double r97506 = a;
        double r97507 = r97500 / r97506;
        double r97508 = r97505 - r97507;
        double r97509 = r97503 * r97508;
        double r97510 = -1.9415295134599817e-260;
        bool r97511 = r97500 <= r97510;
        double r97512 = -r97500;
        double r97513 = r97500 * r97500;
        double r97514 = 4.0;
        double r97515 = r97514 * r97506;
        double r97516 = r97515 * r97504;
        double r97517 = r97513 - r97516;
        double r97518 = sqrt(r97517);
        double r97519 = r97512 + r97518;
        double r97520 = 1.0;
        double r97521 = 2.0;
        double r97522 = r97521 * r97506;
        double r97523 = r97520 / r97522;
        double r97524 = r97519 * r97523;
        double r97525 = 1.7521504885677008e+103;
        bool r97526 = r97500 <= r97525;
        double r97527 = r97521 * r97504;
        double r97528 = r97512 - r97518;
        double r97529 = r97527 / r97528;
        double r97530 = -1.0;
        double r97531 = r97530 * r97505;
        double r97532 = r97526 ? r97529 : r97531;
        double r97533 = r97511 ? r97524 : r97532;
        double r97534 = r97502 ? r97509 : r97533;
        return r97534;
}

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

Original34.3
Target20.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.2975228517561493e+130

    1. Initial program 54.6

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -4.2975228517561493e+130 < b < -1.9415295134599817e-260

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if -1.9415295134599817e-260 < b < 1.7521504885677008e+103

    1. Initial program 32.0

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

    3. Applied flip-+32.1

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

    4. Simplified16.4

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

    6. Applied div-inv16.4

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

    7. Applied associate-/l*21.0

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

    8. Simplified21.0

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

    10. Applied associate-/r*15.4

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

    11. Simplified15.4

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

    12. Taylor expanded around 0 9.0

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

    if 1.7521504885677008e+103 < b

    1. Initial program 59.7

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.941529513459981659526481896938835655376 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.752150488567700802544439975375947245276 \cdot 10^{103}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))