Average Error: 33.0 → 9.9
Time: 2.6m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.139254247068609 \cdot 10^{+140}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.296656918443349 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -9.139254247068609 \cdot 10^{+140}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 5.296656918443349 \cdot 10^{-44}:\\
\;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r7824502 = b;
        double r7824503 = -r7824502;
        double r7824504 = r7824502 * r7824502;
        double r7824505 = 4.0;
        double r7824506 = a;
        double r7824507 = c;
        double r7824508 = r7824506 * r7824507;
        double r7824509 = r7824505 * r7824508;
        double r7824510 = r7824504 - r7824509;
        double r7824511 = sqrt(r7824510);
        double r7824512 = r7824503 + r7824511;
        double r7824513 = 2.0;
        double r7824514 = r7824513 * r7824506;
        double r7824515 = r7824512 / r7824514;
        return r7824515;
}
↓
double f(double a, double b, double c) {
        double r7824516 = b;
        double r7824517 = -9.139254247068609e+140;
        bool r7824518 = r7824516 <= r7824517;
        double r7824519 = c;
        double r7824520 = r7824519 / r7824516;
        double r7824521 = a;
        double r7824522 = r7824516 / r7824521;
        double r7824523 = r7824520 - r7824522;
        double r7824524 = 5.296656918443349e-44;
        bool r7824525 = r7824516 <= r7824524;
        double r7824526 = -4.0;
        double r7824527 = r7824519 * r7824521;
        double r7824528 = r7824516 * r7824516;
        double r7824529 = fma(r7824526, r7824527, r7824528);
        double r7824530 = sqrt(r7824529);
        double r7824531 = r7824530 - r7824516;
        double r7824532 = r7824531 / r7824521;
        double r7824533 = 0.5;
        double r7824534 = r7824532 * r7824533;
        double r7824535 = -r7824519;
        double r7824536 = r7824535 / r7824516;
        double r7824537 = r7824525 ? r7824534 : r7824536;
        double r7824538 = r7824518 ? r7824523 : r7824537;
        return r7824538;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.0
Target20.1
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.139254247068609e+140

    1. Initial program 55.8

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

    2. Simplified55.8

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

    3. Taylor expanded around -inf 1.6

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

    if -9.139254247068609e+140 < b < 5.296656918443349e-44

    1. Initial program 13.4

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

    2. Simplified13.4

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

    4. Applied clear-num13.5

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

    6. Applied associate-/r/13.5

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

    7. Applied *-un-lft-identity13.5

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

    8. Applied times-frac13.5

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

    9. Simplified13.4

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

    10. Simplified13.4

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

    if 5.296656918443349e-44 < 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

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

    3. Taylor expanded around inf 7.4

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

    4. Simplified7.4

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.139254247068609 \cdot 10^{+140}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 5.296656918443349 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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