Average Error: 34.6 → 7.5
Time: 1.1m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -4.718078597954240507360630293401646945929 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.132821374632338820562375169502615703862 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]
\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 -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le -4.718078597954240507360630293401646945929 \cdot 10^{-288}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\

\mathbf{elif}\;b \le 1.132821374632338820562375169502615703862 \cdot 10^{81}:\\
\;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r152487 = b;
        double r152488 = -r152487;
        double r152489 = r152487 * r152487;
        double r152490 = 4.0;
        double r152491 = a;
        double r152492 = r152490 * r152491;
        double r152493 = c;
        double r152494 = r152492 * r152493;
        double r152495 = r152489 - r152494;
        double r152496 = sqrt(r152495);
        double r152497 = r152488 + r152496;
        double r152498 = 2.0;
        double r152499 = r152498 * r152491;
        double r152500 = r152497 / r152499;
        return r152500;
}
↓
double f(double a, double b, double c) {
        double r152501 = b;
        double r152502 = -7.943482039519134e+75;
        bool r152503 = r152501 <= r152502;
        double r152504 = c;
        double r152505 = r152504 / r152501;
        double r152506 = a;
        double r152507 = r152501 / r152506;
        double r152508 = r152505 - r152507;
        double r152509 = 1.0;
        double r152510 = r152508 * r152509;
        double r152511 = -4.7180785979542405e-288;
        bool r152512 = r152501 <= r152511;
        double r152513 = r152501 * r152501;
        double r152514 = 4.0;
        double r152515 = r152514 * r152506;
        double r152516 = r152515 * r152504;
        double r152517 = r152513 - r152516;
        double r152518 = sqrt(r152517);
        double r152519 = -r152501;
        double r152520 = r152518 + r152519;
        double r152521 = 2.0;
        double r152522 = r152506 * r152521;
        double r152523 = r152520 / r152522;
        double r152524 = 1.1328213746323388e+81;
        bool r152525 = r152501 <= r152524;
        double r152526 = r152504 * r152506;
        double r152527 = r152514 * r152526;
        double r152528 = r152513 - r152527;
        double r152529 = sqrt(r152528);
        double r152530 = r152519 - r152529;
        double r152531 = cbrt(r152530);
        double r152532 = r152531 / r152504;
        double r152533 = cbrt(r152532);
        double r152534 = r152506 / r152533;
        double r152535 = r152534 / r152531;
        double r152536 = r152519 - r152518;
        double r152537 = cbrt(r152536);
        double r152538 = r152535 / r152537;
        double r152539 = r152538 / r152506;
        double r152540 = r152533 * r152533;
        double r152541 = r152514 / r152540;
        double r152542 = r152541 / r152521;
        double r152543 = r152539 * r152542;
        double r152544 = -1.0;
        double r152545 = r152504 * r152544;
        double r152546 = r152545 / r152501;
        double r152547 = r152525 ? r152543 : r152546;
        double r152548 = r152512 ? r152523 : r152547;
        double r152549 = r152503 ? r152510 : r152548;
        return r152549;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -7.943482039519134e+75

    1. Initial program 42.7

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

    2. Taylor expanded around -inf 4.2

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

    3. Simplified4.2

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

    if -7.943482039519134e+75 < b < -4.7180785979542405e-288

    1. Initial program 9.3

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

    if -4.7180785979542405e-288 < b < 1.1328213746323388e+81

    1. Initial program 30.8

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

    3. Applied flip-+30.8

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

    4. Simplified16.0

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

    6. Applied add-cube-cbrt16.7

      \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\color{blue}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

    7. Applied associate-/r*16.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

    8. Simplified16.0

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

    10. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{\frac{\frac{4 \cdot a}{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\sqrt[3]{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}}{2 \cdot a}\]

    11. Applied cbrt-prod16.0

      \[\leadsto \frac{\frac{\frac{\frac{4 \cdot a}{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

    12. Applied *-un-lft-identity16.0

      \[\leadsto \frac{\frac{\frac{\frac{4 \cdot a}{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}{\color{blue}{1 \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{\sqrt[3]{1} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    13. Applied add-cube-cbrt16.2

      \[\leadsto \frac{\frac{\frac{\frac{4 \cdot a}{\color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}}{1 \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\sqrt[3]{1} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    14. Applied times-frac16.2

      \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}} \cdot \frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}}{1 \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\sqrt[3]{1} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    15. Applied times-frac15.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}{1} \cdot \frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{\sqrt[3]{1} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    16. Applied times-frac15.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}{1}}{\sqrt[3]{1}} \cdot \frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

    17. Applied times-frac12.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}{1}}{\sqrt[3]{1}}}{2} \cdot \frac{\frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a}}\]

    if 1.1328213746323388e+81 < b

    1. Initial program 59.0

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

    2. Taylor expanded around inf 2.5

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

    3. Simplified2.5

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -4.718078597954240507360630293401646945929 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.132821374632338820562375169502615703862 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))