Average Error: 34.0 → 7.1
Time: 32.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.089942740476039 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1606217053284985 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.9653089193303188 \cdot 10^{+135}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.089942740476039 \cdot 10^{+37}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.1606217053284985 \cdot 10^{-301}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 1.9653089193303188 \cdot 10^{+135}:\\
\;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r15646448 = b;
        double r15646449 = -r15646448;
        double r15646450 = r15646448 * r15646448;
        double r15646451 = 4.0;
        double r15646452 = a;
        double r15646453 = r15646451 * r15646452;
        double r15646454 = c;
        double r15646455 = r15646453 * r15646454;
        double r15646456 = r15646450 - r15646455;
        double r15646457 = sqrt(r15646456);
        double r15646458 = r15646449 + r15646457;
        double r15646459 = 2.0;
        double r15646460 = r15646459 * r15646452;
        double r15646461 = r15646458 / r15646460;
        return r15646461;
}
↓
double f(double a, double b, double c) {
        double r15646462 = b;
        double r15646463 = -5.089942740476039e+37;
        bool r15646464 = r15646462 <= r15646463;
        double r15646465 = c;
        double r15646466 = r15646465 / r15646462;
        double r15646467 = a;
        double r15646468 = r15646462 / r15646467;
        double r15646469 = r15646466 - r15646468;
        double r15646470 = 1.1606217053284985e-301;
        bool r15646471 = r15646462 <= r15646470;
        double r15646472 = r15646462 * r15646462;
        double r15646473 = r15646465 * r15646467;
        double r15646474 = 4.0;
        double r15646475 = r15646473 * r15646474;
        double r15646476 = r15646472 - r15646475;
        double r15646477 = sqrt(r15646476);
        double r15646478 = r15646477 - r15646462;
        double r15646479 = 2.0;
        double r15646480 = r15646467 * r15646479;
        double r15646481 = r15646478 / r15646480;
        double r15646482 = 1.9653089193303188e+135;
        bool r15646483 = r15646462 <= r15646482;
        double r15646484 = -2.0;
        double r15646485 = r15646465 * r15646484;
        double r15646486 = r15646477 + r15646462;
        double r15646487 = r15646485 / r15646486;
        double r15646488 = -r15646466;
        double r15646489 = r15646483 ? r15646487 : r15646488;
        double r15646490 = r15646471 ? r15646481 : r15646489;
        double r15646491 = r15646464 ? r15646469 : r15646490;
        return r15646491;
}

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

Original34.0
Target20.7
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.089942740476039e+37

    1. Initial program 34.9

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

    2. Simplified34.9

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

    4. Applied div-inv35.0

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

    5. Taylor expanded around -inf 6.3

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

    if -5.089942740476039e+37 < b < 1.1606217053284985e-301

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied div-inv10.4

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

    6. Applied un-div-inv10.3

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

    if 1.1606217053284985e-301 < b < 1.9653089193303188e+135

    1. Initial program 34.6

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

    2. Simplified34.6

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

    4. Applied div-inv34.6

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

    6. Applied flip--34.7

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

    7. Applied associate-*l/34.7

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

    8. Simplified14.7

      \[\leadsto \frac{\color{blue}{\frac{0 - \left(c \cdot a\right) \cdot 4}{\frac{a}{\frac{1}{2}}}}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

    9. Taylor expanded around 0 8.3

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

    if 1.9653089193303188e+135 < b

    1. Initial program 61.2

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

    2. Simplified61.2

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

    3. Taylor expanded around inf 2.1

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

    4. Simplified2.1

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.089942740476039 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1606217053284985 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.9653089193303188 \cdot 10^{+135}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))