Average Error: 33.2 → 10.0
Time: 20.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -7.397994825724217 \cdot 10^{+150}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6411408 = b;
        double r6411409 = -r6411408;
        double r6411410 = r6411408 * r6411408;
        double r6411411 = 4.0;
        double r6411412 = a;
        double r6411413 = r6411411 * r6411412;
        double r6411414 = c;
        double r6411415 = r6411413 * r6411414;
        double r6411416 = r6411410 - r6411415;
        double r6411417 = sqrt(r6411416);
        double r6411418 = r6411409 + r6411417;
        double r6411419 = 2.0;
        double r6411420 = r6411419 * r6411412;
        double r6411421 = r6411418 / r6411420;
        return r6411421;
}
↓
double f(double a, double b, double c) {
        double r6411422 = b;
        double r6411423 = -7.397994825724217e+150;
        bool r6411424 = r6411422 <= r6411423;
        double r6411425 = c;
        double r6411426 = r6411425 / r6411422;
        double r6411427 = a;
        double r6411428 = r6411422 / r6411427;
        double r6411429 = r6411426 - r6411428;
        double r6411430 = 2.0;
        double r6411431 = r6411429 * r6411430;
        double r6411432 = r6411431 / r6411430;
        double r6411433 = 1.2158870426682226e-82;
        bool r6411434 = r6411422 <= r6411433;
        double r6411435 = 1.0;
        double r6411436 = r6411422 * r6411422;
        double r6411437 = r6411427 * r6411425;
        double r6411438 = 4.0;
        double r6411439 = r6411437 * r6411438;
        double r6411440 = r6411436 - r6411439;
        double r6411441 = sqrt(r6411440);
        double r6411442 = r6411427 / r6411441;
        double r6411443 = r6411435 / r6411442;
        double r6411444 = r6411443 - r6411428;
        double r6411445 = r6411444 / r6411430;
        double r6411446 = -2.0;
        double r6411447 = r6411446 * r6411426;
        double r6411448 = r6411447 / r6411430;
        double r6411449 = r6411434 ? r6411445 : r6411448;
        double r6411450 = r6411424 ? r6411432 : r6411449;
        return r6411450;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if b < -7.397994825724217e+150

    1. Initial program 59.1

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

    2. Simplified59.1

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -7.397994825724217e+150 < b < 1.2158870426682226e-82

    1. Initial program 11.8

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

    2. Simplified11.7

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

    4. Applied div-sub11.7

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

    6. Applied clear-num11.8

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

    if 1.2158870426682226e-82 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.397994825724217 \cdot 10^{+150}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.2158870426682226 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))