Average Error: 34.5 → 8.9
Time: 15.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.706117685651469092807044052871735370705 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 16056697633982866014018962968901844992:\\ \;\;\;\;\frac{1}{\left(2 \cdot a\right) \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 8.706117685651469092807044052871735370705 \cdot 10^{-130}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

\mathbf{elif}\;b \le 16056697633982866014018962968901844992:\\
\;\;\;\;\frac{1}{\left(2 \cdot a\right) \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\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 r88417 = b;
        double r88418 = -r88417;
        double r88419 = r88417 * r88417;
        double r88420 = 4.0;
        double r88421 = a;
        double r88422 = r88420 * r88421;
        double r88423 = c;
        double r88424 = r88422 * r88423;
        double r88425 = r88419 - r88424;
        double r88426 = sqrt(r88425);
        double r88427 = r88418 + r88426;
        double r88428 = 2.0;
        double r88429 = r88428 * r88421;
        double r88430 = r88427 / r88429;
        return r88430;
}
↓
double f(double a, double b, double c) {
        double r88431 = b;
        double r88432 = -1.8572382657132166e+109;
        bool r88433 = r88431 <= r88432;
        double r88434 = 1.0;
        double r88435 = c;
        double r88436 = r88435 / r88431;
        double r88437 = a;
        double r88438 = r88431 / r88437;
        double r88439 = r88436 - r88438;
        double r88440 = r88434 * r88439;
        double r88441 = 8.706117685651469e-130;
        bool r88442 = r88431 <= r88441;
        double r88443 = 1.0;
        double r88444 = 2.0;
        double r88445 = r88444 * r88437;
        double r88446 = r88431 * r88431;
        double r88447 = 4.0;
        double r88448 = r88447 * r88437;
        double r88449 = r88448 * r88435;
        double r88450 = r88446 - r88449;
        double r88451 = sqrt(r88450);
        double r88452 = r88451 - r88431;
        double r88453 = r88445 / r88452;
        double r88454 = r88443 / r88453;
        double r88455 = 1.6056697633982866e+37;
        bool r88456 = r88431 <= r88455;
        double r88457 = -r88431;
        double r88458 = r88457 - r88451;
        double r88459 = r88458 / r88449;
        double r88460 = r88445 * r88459;
        double r88461 = r88443 / r88460;
        double r88462 = -1.0;
        double r88463 = r88462 * r88436;
        double r88464 = r88456 ? r88461 : r88463;
        double r88465 = r88442 ? r88454 : r88464;
        double r88466 = r88433 ? r88440 : r88465;
        return r88466;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -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. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.8572382657132166e+109 < b < 8.706117685651469e-130

    1. Initial program 11.5

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

    3. Applied clear-num11.6

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

    4. Simplified11.6

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

    if 8.706117685651469e-130 < b < 1.6056697633982866e+37

    1. Initial program 37.3

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

    3. Applied flip-+37.3

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

    4. Simplified16.8

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

    6. Applied clear-num16.9

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

    7. Simplified16.9

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

    9. Applied *-un-lft-identity16.9

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

    10. Applied add-sqr-sqrt16.9

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

    11. Applied times-frac16.9

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

    12. Applied associate-/l*17.0

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

    13. Simplified16.9

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

    if 1.6056697633982866e+37 < b

    1. Initial program 57.1

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

    2. Taylor expanded around inf 4.2

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.706117685651469092807044052871735370705 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 16056697633982866014018962968901844992:\\ \;\;\;\;\frac{1}{\left(2 \cdot a\right) \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

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)))