Average Error: 34.4 → 7.0
Time: 10.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r99428 = b;
        double r99429 = -r99428;
        double r99430 = r99428 * r99428;
        double r99431 = 4.0;
        double r99432 = a;
        double r99433 = c;
        double r99434 = r99432 * r99433;
        double r99435 = r99431 * r99434;
        double r99436 = r99430 - r99435;
        double r99437 = sqrt(r99436);
        double r99438 = r99429 - r99437;
        double r99439 = 2.0;
        double r99440 = r99439 * r99432;
        double r99441 = r99438 / r99440;
        return r99441;
}
↓
double f(double a, double b, double c) {
        double r99442 = b;
        double r99443 = -1.457738542065717e+153;
        bool r99444 = r99442 <= r99443;
        double r99445 = -1.0;
        double r99446 = c;
        double r99447 = r99446 / r99442;
        double r99448 = r99445 * r99447;
        double r99449 = -2.695863739873929e-295;
        bool r99450 = r99442 <= r99449;
        double r99451 = 2.0;
        double r99452 = r99451 * r99446;
        double r99453 = r99442 * r99442;
        double r99454 = 4.0;
        double r99455 = a;
        double r99456 = r99455 * r99446;
        double r99457 = r99454 * r99456;
        double r99458 = r99453 - r99457;
        double r99459 = sqrt(r99458);
        double r99460 = r99459 - r99442;
        double r99461 = r99452 / r99460;
        double r99462 = 1.1912031425131646e+117;
        bool r99463 = r99442 <= r99462;
        double r99464 = 1.0;
        double r99465 = r99451 * r99455;
        double r99466 = -r99442;
        double r99467 = r99466 - r99459;
        double r99468 = r99465 / r99467;
        double r99469 = r99464 / r99468;
        double r99470 = r99442 / r99455;
        double r99471 = r99445 * r99470;
        double r99472 = r99463 ? r99469 : r99471;
        double r99473 = r99450 ? r99461 : r99472;
        double r99474 = r99444 ? r99448 : r99473;
        return r99474;
}

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

Original34.4
Target21.2
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.457738542065717e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.3

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

    if -1.457738542065717e+153 < b < -2.695863739873929e-295

    1. Initial program 35.5

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

    3. Applied flip--35.5

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied div-inv16.6

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

    9. Applied associate-*l/15.1

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

    10. Simplified15.0

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

    11. Taylor expanded around 0 8.3

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

    if -2.695863739873929e-295 < b < 1.1912031425131646e+117

    1. Initial program 9.8

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

    3. Applied clear-num9.9

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

    if 1.1912031425131646e+117 < b

    1. Initial program 50.7

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

    3. Applied flip--63.7

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

    4. Simplified62.7

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

    5. Simplified62.7

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

    6. Taylor expanded around 0 4.0

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))