Average Error: 28.8 → 16.2
Time: 17.8s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 1083.7283358723973:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r677599 = b;
        double r677600 = -r677599;
        double r677601 = r677599 * r677599;
        double r677602 = 4.0;
        double r677603 = a;
        double r677604 = r677602 * r677603;
        double r677605 = c;
        double r677606 = r677604 * r677605;
        double r677607 = r677601 - r677606;
        double r677608 = sqrt(r677607);
        double r677609 = r677600 + r677608;
        double r677610 = 2.0;
        double r677611 = r677610 * r677603;
        double r677612 = r677609 / r677611;
        return r677612;
}
↓
double f(double a, double b, double c) {
        double r677613 = b;
        double r677614 = 1083.7283358723973;
        bool r677615 = r677613 <= r677614;
        double r677616 = c;
        double r677617 = -4.0;
        double r677618 = r677616 * r677617;
        double r677619 = a;
        double r677620 = r677613 * r677613;
        double r677621 = fma(r677618, r677619, r677620);
        double r677622 = sqrt(r677621);
        double r677623 = r677622 * r677621;
        double r677624 = r677620 * r677613;
        double r677625 = r677623 - r677624;
        double r677626 = r677620 + r677621;
        double r677627 = fma(r677613, r677622, r677626);
        double r677628 = r677625 / r677627;
        double r677629 = r677628 / r677619;
        double r677630 = 2.0;
        double r677631 = r677629 / r677630;
        double r677632 = -2.0;
        double r677633 = r677616 / r677613;
        double r677634 = r677632 * r677633;
        double r677635 = r677634 / r677630;
        double r677636 = r677615 ? r677631 : r677635;
        return r677636;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1083.7283358723973

    1. Initial program 17.4

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

    2. Simplified17.3

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

    4. Applied flip3--17.4

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

    5. Simplified16.7

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

    6. Simplified16.7

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

    if 1083.7283358723973 < b

    1. Initial program 36.7

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

    2. Simplified36.6

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

    3. Taylor expanded around inf 15.9

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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)))