Average Error: 32.5 → 9.9
Time: 27.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4266250849096228 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.4266250849096228 \cdot 10^{-56}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2704407 = b;
        double r2704408 = -r2704407;
        double r2704409 = r2704407 * r2704407;
        double r2704410 = 4.0;
        double r2704411 = a;
        double r2704412 = c;
        double r2704413 = r2704411 * r2704412;
        double r2704414 = r2704410 * r2704413;
        double r2704415 = r2704409 - r2704414;
        double r2704416 = sqrt(r2704415);
        double r2704417 = r2704408 - r2704416;
        double r2704418 = 2.0;
        double r2704419 = r2704418 * r2704411;
        double r2704420 = r2704417 / r2704419;
        return r2704420;
}
↓
double f(double a, double b, double c) {
        double r2704421 = b;
        double r2704422 = -1.4266250849096228e-56;
        bool r2704423 = r2704421 <= r2704422;
        double r2704424 = c;
        double r2704425 = r2704424 / r2704421;
        double r2704426 = -r2704425;
        double r2704427 = 2.2373425340727037e+98;
        bool r2704428 = r2704421 <= r2704427;
        double r2704429 = 1.0;
        double r2704430 = 2.0;
        double r2704431 = a;
        double r2704432 = r2704430 * r2704431;
        double r2704433 = -r2704421;
        double r2704434 = r2704421 * r2704421;
        double r2704435 = -4.0;
        double r2704436 = r2704435 * r2704424;
        double r2704437 = r2704431 * r2704436;
        double r2704438 = r2704434 + r2704437;
        double r2704439 = sqrt(r2704438);
        double r2704440 = r2704433 - r2704439;
        double r2704441 = r2704432 / r2704440;
        double r2704442 = r2704429 / r2704441;
        double r2704443 = r2704421 / r2704431;
        double r2704444 = r2704425 - r2704443;
        double r2704445 = r2704428 ? r2704442 : r2704444;
        double r2704446 = r2704423 ? r2704426 : r2704445;
        return r2704446;
}

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

Original32.5
Target20.0
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.4266250849096228e-56

    1. Initial program 52.5

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

    3. Applied sub-neg52.5

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

    4. Simplified52.5

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

    5. Taylor expanded around -inf 8.3

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

    6. Simplified8.3

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

    if -1.4266250849096228e-56 < b < 2.2373425340727037e+98

    1. Initial program 12.7

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

    3. Applied sub-neg12.7

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

    4. Simplified12.8

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

    6. Applied *-un-lft-identity12.8

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

    7. Applied *-un-lft-identity12.8

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

    8. Applied distribute-rgt-neg-in12.8

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

    9. Applied distribute-lft-out--12.8

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

    10. Applied associate-/l*12.9

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

    if 2.2373425340727037e+98 < b

    1. Initial program 43.2

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

    3. Applied sub-neg43.2

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

    4. Simplified43.2

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

    5. Taylor expanded around inf 4.9

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.4266250849096228 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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)))