Average Error: 28.6 → 16.3
Time: 15.4s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6015.720092576997558353468775749206542969:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{1}{c}}}{b}}{2}\\ \end{array}\]
\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 6015.720092576997558353468775749206542969:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{-2}{\frac{1}{c}}}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1602449 = b;
        double r1602450 = -r1602449;
        double r1602451 = r1602449 * r1602449;
        double r1602452 = 4.0;
        double r1602453 = a;
        double r1602454 = r1602452 * r1602453;
        double r1602455 = c;
        double r1602456 = r1602454 * r1602455;
        double r1602457 = r1602451 - r1602456;
        double r1602458 = sqrt(r1602457);
        double r1602459 = r1602450 + r1602458;
        double r1602460 = 2.0;
        double r1602461 = r1602460 * r1602453;
        double r1602462 = r1602459 / r1602461;
        return r1602462;
}
↓
double f(double a, double b, double c) {
        double r1602463 = b;
        double r1602464 = 6015.720092576998;
        bool r1602465 = r1602463 <= r1602464;
        double r1602466 = r1602463 * r1602463;
        double r1602467 = a;
        double r1602468 = 4.0;
        double r1602469 = r1602467 * r1602468;
        double r1602470 = c;
        double r1602471 = r1602469 * r1602470;
        double r1602472 = r1602466 - r1602471;
        double r1602473 = sqrt(r1602472);
        double r1602474 = r1602472 * r1602473;
        double r1602475 = r1602466 * r1602463;
        double r1602476 = r1602474 - r1602475;
        double r1602477 = r1602463 * r1602473;
        double r1602478 = r1602477 + r1602466;
        double r1602479 = r1602472 + r1602478;
        double r1602480 = r1602476 / r1602479;
        double r1602481 = r1602480 / r1602467;
        double r1602482 = 2.0;
        double r1602483 = r1602481 / r1602482;
        double r1602484 = -2.0;
        double r1602485 = 1.0;
        double r1602486 = r1602485 / r1602470;
        double r1602487 = r1602484 / r1602486;
        double r1602488 = r1602487 / r1602463;
        double r1602489 = r1602488 / r1602482;
        double r1602490 = r1602465 ? r1602483 : r1602489;
        return r1602490;
}

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 2 regimes

  2. if b < 6015.720092576998

    1. Initial program 18.6

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

    2. Simplified18.6

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

    4. Applied flip3--18.6

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

    5. Simplified17.9

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

    6. Simplified17.9

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

    if 6015.720092576998 < b

    1. Initial program 38.2

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

    2. Simplified38.2

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

    3. Taylor expanded around inf 14.9

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a}}{2}\]
    4. Using strategy rm

    5. Applied associate-*r/14.9

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

    6. Applied associate-/l/14.9

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

    8. Applied associate-/r*14.8

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

    9. Simplified14.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{-2}{\frac{1}{c}}}}{b}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6015.720092576997558353468775749206542969:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{1}{c}}}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))