Average Error: 32.9 → 10.1
Time: 14.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2769456 = b;
        double r2769457 = -r2769456;
        double r2769458 = r2769456 * r2769456;
        double r2769459 = 4.0;
        double r2769460 = a;
        double r2769461 = r2769459 * r2769460;
        double r2769462 = c;
        double r2769463 = r2769461 * r2769462;
        double r2769464 = r2769458 - r2769463;
        double r2769465 = sqrt(r2769464);
        double r2769466 = r2769457 + r2769465;
        double r2769467 = 2.0;
        double r2769468 = r2769467 * r2769460;
        double r2769469 = r2769466 / r2769468;
        return r2769469;
}
↓
double f(double a, double b, double c) {
        double r2769470 = b;
        double r2769471 = -1.7512236628315378e+131;
        bool r2769472 = r2769470 <= r2769471;
        double r2769473 = c;
        double r2769474 = r2769473 / r2769470;
        double r2769475 = a;
        double r2769476 = r2769470 / r2769475;
        double r2769477 = r2769474 - r2769476;
        double r2769478 = 2.0;
        double r2769479 = r2769477 * r2769478;
        double r2769480 = r2769479 / r2769478;
        double r2769481 = 1.489031291672483e-98;
        bool r2769482 = r2769470 <= r2769481;
        double r2769483 = r2769470 * r2769470;
        double r2769484 = 4.0;
        double r2769485 = r2769484 * r2769473;
        double r2769486 = r2769485 * r2769475;
        double r2769487 = r2769483 - r2769486;
        double r2769488 = sqrt(r2769487);
        double r2769489 = r2769488 - r2769470;
        double r2769490 = r2769489 / r2769475;
        double r2769491 = r2769490 / r2769478;
        double r2769492 = -2.0;
        double r2769493 = r2769492 * r2769474;
        double r2769494 = r2769493 / r2769478;
        double r2769495 = r2769482 ? r2769491 : r2769494;
        double r2769496 = r2769472 ? r2769480 : r2769495;
        return r2769496;
}

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
Target20.4
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7512236628315378e+131

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around -inf 3.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified3.0

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

    if -1.7512236628315378e+131 < b < 1.489031291672483e-98

    1. Initial program 11.5

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

    2. Simplified11.6

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

    4. Applied *-un-lft-identity11.6

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

    5. Applied associate-/r*11.6

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

    6. Simplified11.6

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

    if 1.489031291672483e-98 < b

    1. Initial program 51.5

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

    2. Simplified51.5

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

    3. Taylor expanded around inf 10.7

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7512236628315378 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.489031291672483 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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