Average Error: 34.1 → 8.6
Time: 7.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.28731338523231927980681449927859365128 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.914307195398066377772349987498596241276 \cdot 10^{-230}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.314242005915538169906364876769500015544 \cdot 10^{99}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.28731338523231927980681449927859365128 \cdot 10^{78}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.914307195398066377772349987498596241276 \cdot 10^{-230}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.314242005915538169906364876769500015544 \cdot 10^{99}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r104485 = b;
        double r104486 = -r104485;
        double r104487 = r104485 * r104485;
        double r104488 = 4.0;
        double r104489 = a;
        double r104490 = c;
        double r104491 = r104489 * r104490;
        double r104492 = r104488 * r104491;
        double r104493 = r104487 - r104492;
        double r104494 = sqrt(r104493);
        double r104495 = r104486 + r104494;
        double r104496 = 2.0;
        double r104497 = r104496 * r104489;
        double r104498 = r104495 / r104497;
        return r104498;
}
↓
double f(double a, double b, double c) {
        double r104499 = b;
        double r104500 = -4.287313385232319e+78;
        bool r104501 = r104499 <= r104500;
        double r104502 = 1.0;
        double r104503 = c;
        double r104504 = r104503 / r104499;
        double r104505 = a;
        double r104506 = r104499 / r104505;
        double r104507 = r104504 - r104506;
        double r104508 = r104502 * r104507;
        double r104509 = -3.9143071953980664e-230;
        bool r104510 = r104499 <= r104509;
        double r104511 = -r104499;
        double r104512 = r104499 * r104499;
        double r104513 = 4.0;
        double r104514 = r104505 * r104503;
        double r104515 = r104513 * r104514;
        double r104516 = r104512 - r104515;
        double r104517 = sqrt(r104516);
        double r104518 = sqrt(r104517);
        double r104519 = r104518 * r104518;
        double r104520 = r104511 + r104519;
        double r104521 = 2.0;
        double r104522 = r104521 * r104505;
        double r104523 = r104520 / r104522;
        double r104524 = 3.314242005915538e+99;
        bool r104525 = r104499 <= r104524;
        double r104526 = r104515 / r104522;
        double r104527 = r104511 - r104517;
        double r104528 = r104526 / r104527;
        double r104529 = -1.0;
        double r104530 = r104529 * r104504;
        double r104531 = r104525 ? r104528 : r104530;
        double r104532 = r104510 ? r104523 : r104531;
        double r104533 = r104501 ? r104508 : r104532;
        return r104533;
}

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

Original34.1
Target20.7
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.287313385232319e+78

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 4.2

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

    3. Simplified4.2

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

    if -4.287313385232319e+78 < b < -3.9143071953980664e-230

    1. Initial program 7.7

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

    3. Applied add-sqr-sqrt7.7

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

    4. Applied sqrt-prod7.9

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

    if -3.9143071953980664e-230 < b < 3.314242005915538e+99

    1. Initial program 30.8

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

    3. Applied flip-+30.8

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

    4. Simplified16.4

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

    6. Applied div-inv16.4

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

    8. Applied associate-*l/15.6

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

    9. Simplified15.5

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

    if 3.314242005915538e+99 < b

    1. Initial program 59.1

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

    2. Taylor expanded around inf 2.6

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.28731338523231927980681449927859365128 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.914307195398066377772349987498596241276 \cdot 10^{-230}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.314242005915538169906364876769500015544 \cdot 10^{99}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +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)))