Average Error: 34.4 → 10.6
Time: 21.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot \frac{a}{b}, 2, b \cdot -2\right)}{a}}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot 4\right) \cdot \left(-c\right)\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot \frac{a}{b}, 2, b \cdot -2\right)}{a}}{2}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot 4\right) \cdot \left(-c\right)\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4460484 = b;
        double r4460485 = -r4460484;
        double r4460486 = r4460484 * r4460484;
        double r4460487 = 4.0;
        double r4460488 = a;
        double r4460489 = c;
        double r4460490 = r4460488 * r4460489;
        double r4460491 = r4460487 * r4460490;
        double r4460492 = r4460486 - r4460491;
        double r4460493 = sqrt(r4460492);
        double r4460494 = r4460485 + r4460493;
        double r4460495 = 2.0;
        double r4460496 = r4460495 * r4460488;
        double r4460497 = r4460494 / r4460496;
        return r4460497;
}
↓
double f(double a, double b, double c) {
        double r4460498 = b;
        double r4460499 = -2.221067196710922e+149;
        bool r4460500 = r4460498 <= r4460499;
        double r4460501 = c;
        double r4460502 = a;
        double r4460503 = r4460502 / r4460498;
        double r4460504 = r4460501 * r4460503;
        double r4460505 = 2.0;
        double r4460506 = -2.0;
        double r4460507 = r4460498 * r4460506;
        double r4460508 = fma(r4460504, r4460505, r4460507);
        double r4460509 = r4460508 / r4460502;
        double r4460510 = r4460509 / r4460505;
        double r4460511 = 2.8983489306952693e-35;
        bool r4460512 = r4460498 <= r4460511;
        double r4460513 = 4.0;
        double r4460514 = r4460502 * r4460513;
        double r4460515 = -r4460501;
        double r4460516 = r4460514 * r4460515;
        double r4460517 = fma(r4460498, r4460498, r4460516);
        double r4460518 = sqrt(r4460517);
        double r4460519 = r4460518 - r4460498;
        double r4460520 = r4460519 / r4460502;
        double r4460521 = r4460520 / r4460505;
        double r4460522 = -2.0;
        double r4460523 = r4460501 / r4460498;
        double r4460524 = r4460522 * r4460523;
        double r4460525 = r4460524 / r4460505;
        double r4460526 = r4460512 ? r4460521 : r4460525;
        double r4460527 = r4460500 ? r4460510 : r4460526;
        return r4460527;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.5
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    4. Applied fma-neg62.3

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

    5. Taylor expanded around -inf 11.0

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

    6. Simplified2.9

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

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.6

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

    2. Simplified14.6

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

    4. Applied fma-neg14.6

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot \frac{a}{b}, 2, b \cdot -2\right)}{a}}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot 4\right) \cdot \left(-c\right)\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))