Average Error: 33.8 → 9.8
Time: 13.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.0569776426586135 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -7.93152454634661985 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.0569776426586135 \cdot 10^{-106}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\frac{a}{\frac{1}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r100415 = b;
        double r100416 = -r100415;
        double r100417 = r100415 * r100415;
        double r100418 = 4.0;
        double r100419 = a;
        double r100420 = c;
        double r100421 = r100419 * r100420;
        double r100422 = r100418 * r100421;
        double r100423 = r100417 - r100422;
        double r100424 = sqrt(r100423);
        double r100425 = r100416 + r100424;
        double r100426 = 2.0;
        double r100427 = r100426 * r100419;
        double r100428 = r100425 / r100427;
        return r100428;
}
↓
double f(double a, double b, double c) {
        double r100429 = b;
        double r100430 = -7.93152454634662e+153;
        bool r100431 = r100429 <= r100430;
        double r100432 = 1.0;
        double r100433 = c;
        double r100434 = r100433 / r100429;
        double r100435 = a;
        double r100436 = r100429 / r100435;
        double r100437 = r100434 - r100436;
        double r100438 = r100432 * r100437;
        double r100439 = 2.0569776426586135e-106;
        bool r100440 = r100429 <= r100439;
        double r100441 = r100429 * r100429;
        double r100442 = 4.0;
        double r100443 = r100435 * r100433;
        double r100444 = r100442 * r100443;
        double r100445 = r100441 - r100444;
        double r100446 = sqrt(r100445);
        double r100447 = r100446 - r100429;
        double r100448 = 1.0;
        double r100449 = 2.0;
        double r100450 = r100448 / r100449;
        double r100451 = r100435 / r100450;
        double r100452 = r100447 / r100451;
        double r100453 = -1.0;
        double r100454 = r100453 * r100434;
        double r100455 = r100440 ? r100452 : r100454;
        double r100456 = r100431 ? r100438 : r100455;
        return r100456;
}

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

Original33.8
Target21.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.93152454634662e+153

    1. Initial program 63.8

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

    2. Simplified63.8

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

    3. Taylor expanded around -inf 1.9

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

    4. Simplified1.9

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

    if -7.93152454634662e+153 < b < 2.0569776426586135e-106

    1. Initial program 11.2

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

    2. Simplified11.2

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

    4. Applied *-un-lft-identity11.2

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

    5. Applied *-un-lft-identity11.2

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

    6. Applied times-frac11.2

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

    7. Applied associate-/l*11.3

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

    9. Applied div-inv11.3

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

    10. Applied *-un-lft-identity11.3

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

    11. Applied times-frac11.4

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

    12. Applied associate-/r*11.3

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

    13. Simplified11.2

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

    if 2.0569776426586135e-106 < b

    1. Initial program 52.0

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

    2. Simplified52.0

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

    3. Taylor expanded around inf 10.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.0569776426586135 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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