Average Error: 28.8 → 16.3
Time: 41.5s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 792.131058127984:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 792.131058127984:\\
\;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5622584 = b;
        double r5622585 = -r5622584;
        double r5622586 = r5622584 * r5622584;
        double r5622587 = 4.0;
        double r5622588 = a;
        double r5622589 = r5622587 * r5622588;
        double r5622590 = c;
        double r5622591 = r5622589 * r5622590;
        double r5622592 = r5622586 - r5622591;
        double r5622593 = sqrt(r5622592);
        double r5622594 = r5622585 + r5622593;
        double r5622595 = 2.0;
        double r5622596 = r5622595 * r5622588;
        double r5622597 = r5622594 / r5622596;
        return r5622597;
}
↓
double f(double a, double b, double c) {
        double r5622598 = b;
        double r5622599 = 792.131058127984;
        bool r5622600 = r5622598 <= r5622599;
        double r5622601 = r5622598 * r5622598;
        double r5622602 = 4.0;
        double r5622603 = c;
        double r5622604 = a;
        double r5622605 = r5622603 * r5622604;
        double r5622606 = r5622602 * r5622605;
        double r5622607 = r5622601 - r5622606;
        double r5622608 = sqrt(r5622607);
        double r5622609 = r5622607 * r5622608;
        double r5622610 = r5622601 * r5622598;
        double r5622611 = r5622609 - r5622610;
        double r5622612 = 2.0;
        double r5622613 = r5622612 * r5622604;
        double r5622614 = r5622598 * r5622608;
        double r5622615 = r5622601 + r5622614;
        double r5622616 = r5622608 * r5622608;
        double r5622617 = r5622615 + r5622616;
        double r5622618 = r5622613 * r5622617;
        double r5622619 = r5622611 / r5622618;
        double r5622620 = r5622603 / r5622598;
        double r5622621 = -r5622620;
        double r5622622 = r5622600 ? r5622619 : r5622621;
        return r5622622;
}

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 < 792.131058127984

    1. Initial program 16.9

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

    2. Simplified16.9

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

    4. Applied flip3--17.0

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

    5. Applied associate-/l/17.0

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

    6. Simplified16.3

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

    if 792.131058127984 < b

    1. Initial program 36.4

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

    2. Simplified36.4

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

    3. Taylor expanded around inf 16.3

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

    5. Applied *-un-lft-identity16.3

      \[\leadsto \frac{-2 \cdot \frac{a \cdot c}{\color{blue}{1 \cdot b}}}{2 \cdot a}\]

    6. Applied times-frac16.2

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

    7. Simplified16.2

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

    8. Taylor expanded around -inf 16.2

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

    9. Simplified16.2

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 792.131058127984:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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 a) c)))) (* 2 a)))