Average Error: 34.2 → 8.3
Time: 14.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.829508521752968200470883584177554571598 \cdot 10^{-234}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{2}\\ \mathbf{elif}\;b \le 2.080207944760409970231511154919136216465 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{4 \cdot a}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.829508521752968200470883584177554571598 \cdot 10^{-234}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{2}\\

\mathbf{elif}\;b \le 2.080207944760409970231511154919136216465 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{4 \cdot a}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r767411 = b;
        double r767412 = -r767411;
        double r767413 = r767411 * r767411;
        double r767414 = 4.0;
        double r767415 = a;
        double r767416 = r767414 * r767415;
        double r767417 = c;
        double r767418 = r767416 * r767417;
        double r767419 = r767413 - r767418;
        double r767420 = sqrt(r767419);
        double r767421 = r767412 + r767420;
        double r767422 = 2.0;
        double r767423 = r767422 * r767415;
        double r767424 = r767421 / r767423;
        return r767424;
}
↓
double f(double a, double b, double c) {
        double r767425 = b;
        double r767426 = -3.450829996567048e+138;
        bool r767427 = r767425 <= r767426;
        double r767428 = c;
        double r767429 = r767428 / r767425;
        double r767430 = a;
        double r767431 = r767425 / r767430;
        double r767432 = r767429 - r767431;
        double r767433 = 1.0;
        double r767434 = r767432 * r767433;
        double r767435 = 2.829508521752968e-234;
        bool r767436 = r767425 <= r767435;
        double r767437 = r767425 * r767425;
        double r767438 = 4.0;
        double r767439 = r767438 * r767430;
        double r767440 = r767439 * r767428;
        double r767441 = r767437 - r767440;
        double r767442 = sqrt(r767441);
        double r767443 = -r767425;
        double r767444 = r767442 + r767443;
        double r767445 = 1.0;
        double r767446 = r767445 / r767430;
        double r767447 = 2.0;
        double r767448 = r767446 / r767447;
        double r767449 = r767444 * r767448;
        double r767450 = 2.08020794476041e-46;
        bool r767451 = r767425 <= r767450;
        double r767452 = r767439 / r767447;
        double r767453 = r767428 * r767430;
        double r767454 = r767453 * r767438;
        double r767455 = r767437 - r767454;
        double r767456 = sqrt(r767455);
        double r767457 = r767443 - r767456;
        double r767458 = r767428 / r767457;
        double r767459 = r767452 * r767458;
        double r767460 = r767459 / r767430;
        double r767461 = -1.0;
        double r767462 = r767429 * r767461;
        double r767463 = r767451 ? r767460 : r767462;
        double r767464 = r767436 ? r767449 : r767463;
        double r767465 = r767427 ? r767434 : r767464;
        return r767465;
}

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

Original34.2
Target21.0
Herbie8.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.450829996567048e+138

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -3.450829996567048e+138 < b < 2.829508521752968e-234

    1. Initial program 8.7

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

    3. Applied div-inv8.9

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

    4. Simplified8.9

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

    if 2.829508521752968e-234 < b < 2.08020794476041e-46

    1. Initial program 25.7

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

    3. Applied flip-+25.7

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

    4. Simplified19.0

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied associate-/r*19.0

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

    7. Simplified14.7

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

    if 2.08020794476041e-46 < b

    1. Initial program 54.2

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

    2. Taylor expanded around inf 7.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.829508521752968200470883584177554571598 \cdot 10^{-234}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{2}\\ \mathbf{elif}\;b \le 2.080207944760409970231511154919136216465 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{4 \cdot a}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))