Average Error: 33.2 → 10.0
Time: 48.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.896735429482593 \cdot 10^{+46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1815645801505244 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -4.896735429482593 \cdot 10^{+46}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.1815645801505244 \cdot 10^{-79}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right) \cdot \frac{1}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r7987398 = b;
        double r7987399 = -r7987398;
        double r7987400 = r7987398 * r7987398;
        double r7987401 = 4.0;
        double r7987402 = a;
        double r7987403 = r7987401 * r7987402;
        double r7987404 = c;
        double r7987405 = r7987403 * r7987404;
        double r7987406 = r7987400 - r7987405;
        double r7987407 = sqrt(r7987406);
        double r7987408 = r7987399 + r7987407;
        double r7987409 = 2.0;
        double r7987410 = r7987409 * r7987402;
        double r7987411 = r7987408 / r7987410;
        return r7987411;
}
↓
double f(double a, double b, double c) {
        double r7987412 = b;
        double r7987413 = -4.896735429482593e+46;
        bool r7987414 = r7987412 <= r7987413;
        double r7987415 = c;
        double r7987416 = r7987415 / r7987412;
        double r7987417 = a;
        double r7987418 = r7987412 / r7987417;
        double r7987419 = r7987416 - r7987418;
        double r7987420 = 1.1815645801505244e-79;
        bool r7987421 = r7987412 <= r7987420;
        double r7987422 = r7987412 * r7987412;
        double r7987423 = r7987415 * r7987417;
        double r7987424 = 4.0;
        double r7987425 = r7987423 * r7987424;
        double r7987426 = r7987422 - r7987425;
        double r7987427 = sqrt(r7987426);
        double r7987428 = r7987427 - r7987412;
        double r7987429 = 1.0;
        double r7987430 = 2.0;
        double r7987431 = r7987417 * r7987430;
        double r7987432 = r7987429 / r7987431;
        double r7987433 = r7987428 * r7987432;
        double r7987434 = -r7987416;
        double r7987435 = r7987421 ? r7987433 : r7987434;
        double r7987436 = r7987414 ? r7987419 : r7987435;
        return r7987436;
}

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

Original33.2
Target20.4
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.896735429482593e+46

    1. Initial program 34.6

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

    2. Simplified34.5

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

    3. Taylor expanded around -inf 5.4

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

    if -4.896735429482593e+46 < b < 1.1815645801505244e-79

    1. Initial program 12.7

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

    2. Simplified12.8

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

    4. Applied div-inv12.9

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

    if 1.1815645801505244e-79 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    4. Applied div-inv52.3

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

    5. Taylor expanded around inf 9.6

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

    6. Simplified9.6

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.896735429482593 \cdot 10^{+46}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1815645801505244 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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)))