Average Error: 33.6 → 10.0
Time: 22.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -3.2092322739463293 \cdot 10^{-86}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\
\;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)}\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2673522 = b;
        double r2673523 = -r2673522;
        double r2673524 = r2673522 * r2673522;
        double r2673525 = 4.0;
        double r2673526 = a;
        double r2673527 = c;
        double r2673528 = r2673526 * r2673527;
        double r2673529 = r2673525 * r2673528;
        double r2673530 = r2673524 - r2673529;
        double r2673531 = sqrt(r2673530);
        double r2673532 = r2673523 - r2673531;
        double r2673533 = 2.0;
        double r2673534 = r2673533 * r2673526;
        double r2673535 = r2673532 / r2673534;
        return r2673535;
}
↓
double f(double a, double b, double c) {
        double r2673536 = b;
        double r2673537 = -3.2092322739463293e-86;
        bool r2673538 = r2673536 <= r2673537;
        double r2673539 = c;
        double r2673540 = r2673539 / r2673536;
        double r2673541 = -r2673540;
        double r2673542 = 2.891777552454845e+74;
        bool r2673543 = r2673536 <= r2673542;
        double r2673544 = -r2673536;
        double r2673545 = a;
        double r2673546 = r2673539 * r2673545;
        double r2673547 = -4.0;
        double r2673548 = r2673546 * r2673547;
        double r2673549 = fma(r2673536, r2673536, r2673548);
        double r2673550 = sqrt(r2673549);
        double r2673551 = r2673544 - r2673550;
        double r2673552 = 0.5;
        double r2673553 = r2673551 * r2673552;
        double r2673554 = r2673553 / r2673545;
        double r2673555 = r2673544 / r2673545;
        double r2673556 = r2673543 ? r2673554 : r2673555;
        double r2673557 = r2673538 ? r2673541 : r2673556;
        return r2673557;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.7
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.2092322739463293e-86

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 9.4

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

    3. Simplified9.4

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

    if -3.2092322739463293e-86 < b < 2.891777552454845e+74

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    4. Simplified13.2

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

    6. Applied associate-*r/13.1

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

    7. Simplified13.1

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

    if 2.891777552454845e+74 < b

    1. Initial program 38.9

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

    3. Applied *-un-lft-identity38.9

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

    4. Applied associate-/l*39.0

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

    5. Simplified39.0

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

    6. Taylor expanded around 0 4.4

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

    7. Simplified4.4

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.2092322739463293 \cdot 10^{-86}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.891777552454845 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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)))