Average Error: 28.6 → 16.5
Time: 15.8s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 916.8232025518482:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 916.8232025518482:\\
\;\;\;\;\frac{\frac{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}\right)}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1294578 = b;
        double r1294579 = -r1294578;
        double r1294580 = r1294578 * r1294578;
        double r1294581 = 4.0;
        double r1294582 = a;
        double r1294583 = r1294581 * r1294582;
        double r1294584 = c;
        double r1294585 = r1294583 * r1294584;
        double r1294586 = r1294580 - r1294585;
        double r1294587 = sqrt(r1294586);
        double r1294588 = r1294579 + r1294587;
        double r1294589 = 2.0;
        double r1294590 = r1294589 * r1294582;
        double r1294591 = r1294588 / r1294590;
        return r1294591;
}
↓
double f(double a, double b, double c) {
        double r1294592 = b;
        double r1294593 = 916.8232025518482;
        bool r1294594 = r1294592 <= r1294593;
        double r1294595 = a;
        double r1294596 = -4.0;
        double r1294597 = r1294595 * r1294596;
        double r1294598 = c;
        double r1294599 = r1294597 * r1294598;
        double r1294600 = r1294592 * r1294592;
        double r1294601 = r1294599 + r1294600;
        double r1294602 = sqrt(r1294601);
        double r1294603 = r1294601 * r1294602;
        double r1294604 = r1294592 * r1294600;
        double r1294605 = r1294603 - r1294604;
        double r1294606 = r1294592 * r1294602;
        double r1294607 = r1294600 + r1294606;
        double r1294608 = r1294601 + r1294607;
        double r1294609 = r1294605 / r1294608;
        double r1294610 = 2.0;
        double r1294611 = r1294595 * r1294610;
        double r1294612 = r1294609 / r1294611;
        double r1294613 = r1294598 / r1294592;
        double r1294614 = -r1294613;
        double r1294615 = r1294594 ? r1294612 : r1294614;
        return r1294615;
}

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 < 916.8232025518482

    1. Initial program 17.3

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

    2. Simplified17.3

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

    4. Applied flip3--17.5

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

    5. Simplified16.7

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

    6. Simplified16.7

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

    if 916.8232025518482 < b

    1. Initial program 36.0

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

    2. Simplified36.0

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

    3. Taylor expanded around inf 16.4

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

    4. Simplified16.4

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 916.8232025518482:\\ \;\;\;\;\frac{\frac{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(a \cdot -4\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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)))