Average Error: 32.9 → 6.8
Time: 41.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.515406138267436 \cdot 10^{+130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.320492610336173 \cdot 10^{-222}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 5.000815192005961 \cdot 10^{+134}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.515406138267436 \cdot 10^{+130}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -4.320492610336173 \cdot 10^{-222}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\

\mathbf{elif}\;b \le 5.000815192005961 \cdot 10^{+134}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r6982486 = b;
        double r6982487 = -r6982486;
        double r6982488 = r6982486 * r6982486;
        double r6982489 = 4.0;
        double r6982490 = a;
        double r6982491 = c;
        double r6982492 = r6982490 * r6982491;
        double r6982493 = r6982489 * r6982492;
        double r6982494 = r6982488 - r6982493;
        double r6982495 = sqrt(r6982494);
        double r6982496 = r6982487 - r6982495;
        double r6982497 = 2.0;
        double r6982498 = r6982497 * r6982490;
        double r6982499 = r6982496 / r6982498;
        return r6982499;
}
↓
double f(double a, double b, double c) {
        double r6982500 = b;
        double r6982501 = -1.515406138267436e+130;
        bool r6982502 = r6982500 <= r6982501;
        double r6982503 = c;
        double r6982504 = r6982503 / r6982500;
        double r6982505 = -r6982504;
        double r6982506 = -4.320492610336173e-222;
        bool r6982507 = r6982500 <= r6982506;
        double r6982508 = 2.0;
        double r6982509 = r6982508 * r6982503;
        double r6982510 = r6982500 * r6982500;
        double r6982511 = a;
        double r6982512 = -4.0;
        double r6982513 = r6982512 * r6982503;
        double r6982514 = r6982511 * r6982513;
        double r6982515 = r6982510 + r6982514;
        double r6982516 = sqrt(r6982515);
        double r6982517 = -r6982500;
        double r6982518 = r6982516 + r6982517;
        double r6982519 = r6982509 / r6982518;
        double r6982520 = 5.000815192005961e+134;
        bool r6982521 = r6982500 <= r6982520;
        double r6982522 = r6982517 - r6982516;
        double r6982523 = r6982511 * r6982508;
        double r6982524 = r6982522 / r6982523;
        double r6982525 = r6982500 / r6982511;
        double r6982526 = r6982504 - r6982525;
        double r6982527 = r6982521 ? r6982524 : r6982526;
        double r6982528 = r6982507 ? r6982519 : r6982527;
        double r6982529 = r6982502 ? r6982505 : r6982528;
        return r6982529;
}

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

Original32.9
Target20.6
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.515406138267436e+130

    1. Initial program 60.2

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

    2. Taylor expanded around -inf 60.2

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

    3. Simplified60.2

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

    4. Taylor expanded around -inf 2.3

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

    5. Simplified2.3

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

    if -1.515406138267436e+130 < b < -4.320492610336173e-222

    1. Initial program 36.2

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

    2. Taylor expanded around -inf 36.2

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

    3. Simplified36.2

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

    5. Applied div-inv36.2

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

    7. Applied flip--36.3

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

    8. Applied associate-*l/36.3

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

    9. Simplified14.6

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

    10. Taylor expanded around 0 7.2

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

    if -4.320492610336173e-222 < b < 5.000815192005961e+134

    1. Initial program 9.7

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

    2. Taylor expanded around -inf 9.7

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

    3. Simplified9.8

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

    if 5.000815192005961e+134 < b

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 53.5

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

    3. Simplified53.5

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

    4. Taylor expanded around inf 3.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.515406138267436 \cdot 10^{+130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.320492610336173 \cdot 10^{-222}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 5.000815192005961 \cdot 10^{+134}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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