Average Error: 32.6 → 9.7
Time: 28.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2985514 = b;
        double r2985515 = -r2985514;
        double r2985516 = r2985514 * r2985514;
        double r2985517 = 4.0;
        double r2985518 = a;
        double r2985519 = c;
        double r2985520 = r2985518 * r2985519;
        double r2985521 = r2985517 * r2985520;
        double r2985522 = r2985516 - r2985521;
        double r2985523 = sqrt(r2985522);
        double r2985524 = r2985515 - r2985523;
        double r2985525 = 2.0;
        double r2985526 = r2985525 * r2985518;
        double r2985527 = r2985524 / r2985526;
        return r2985527;
}
↓
double f(double a, double b, double c) {
        double r2985528 = b;
        double r2985529 = -6.90131991727783e-39;
        bool r2985530 = r2985528 <= r2985529;
        double r2985531 = c;
        double r2985532 = r2985531 / r2985528;
        double r2985533 = -r2985532;
        double r2985534 = 4.012768074517757e+87;
        bool r2985535 = r2985528 <= r2985534;
        double r2985536 = -r2985528;
        double r2985537 = r2985528 * r2985528;
        double r2985538 = a;
        double r2985539 = r2985531 * r2985538;
        double r2985540 = 4.0;
        double r2985541 = r2985539 * r2985540;
        double r2985542 = r2985537 - r2985541;
        double r2985543 = sqrt(r2985542);
        double r2985544 = r2985536 - r2985543;
        double r2985545 = 0.5;
        double r2985546 = r2985545 / r2985538;
        double r2985547 = r2985544 * r2985546;
        double r2985548 = r2985536 / r2985538;
        double r2985549 = r2985535 ? r2985547 : r2985548;
        double r2985550 = r2985530 ? r2985533 : r2985549;
        return r2985550;
}

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

Target

Original32.6
Target20.0
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.90131991727783e-39

    1. Initial program 53.0

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

    3. Applied div-inv53.1

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

    4. Simplified53.1

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

    5. Taylor expanded around -inf 7.9

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

    6. Simplified7.9

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

    if -6.90131991727783e-39 < b < 4.012768074517757e+87

    1. Initial program 13.2

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

    3. Applied div-inv13.3

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

    4. Simplified13.3

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

    if 4.012768074517757e+87 < b

    1. Initial program 41.3

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

    3. Applied *-un-lft-identity41.3

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

    4. Applied *-un-lft-identity41.3

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

    5. Applied distribute-rgt-neg-in41.3

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

    6. Applied distribute-lft-out--41.3

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

    7. Applied associate-/l*41.3

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

    8. Simplified41.3

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

    9. Taylor expanded around 0 3.3

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

    10. Simplified3.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))