Average Error: 33.6 → 10.2
Time: 12.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r117522 = b;
        double r117523 = -r117522;
        double r117524 = r117522 * r117522;
        double r117525 = 4.0;
        double r117526 = a;
        double r117527 = r117525 * r117526;
        double r117528 = c;
        double r117529 = r117527 * r117528;
        double r117530 = r117524 - r117529;
        double r117531 = sqrt(r117530);
        double r117532 = r117523 + r117531;
        double r117533 = 2.0;
        double r117534 = r117533 * r117526;
        double r117535 = r117532 / r117534;
        return r117535;
}
↓
double f(double a, double b, double c) {
        double r117536 = b;
        double r117537 = -1.2609617020890706e+118;
        bool r117538 = r117536 <= r117537;
        double r117539 = 1.0;
        double r117540 = c;
        double r117541 = r117540 / r117536;
        double r117542 = a;
        double r117543 = r117536 / r117542;
        double r117544 = r117541 - r117543;
        double r117545 = r117539 * r117544;
        double r117546 = 5.81843322574321e-115;
        bool r117547 = r117536 <= r117546;
        double r117548 = -r117536;
        double r117549 = r117536 * r117536;
        double r117550 = 4.0;
        double r117551 = r117550 * r117542;
        double r117552 = r117551 * r117540;
        double r117553 = r117549 - r117552;
        double r117554 = sqrt(r117553);
        double r117555 = r117548 + r117554;
        double r117556 = 2.0;
        double r117557 = r117556 * r117542;
        double r117558 = r117555 / r117557;
        double r117559 = -1.0;
        double r117560 = r117559 * r117541;
        double r117561 = r117547 ? r117558 : r117560;
        double r117562 = r117538 ? r117545 : r117561;
        return r117562;
}

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

Original33.6
Target20.5
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2609617020890706e+118

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 2.7

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

    3. Simplified2.7

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

    if -1.2609617020890706e+118 < b < 5.81843322574321e-115

    1. Initial program 11.5

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

    if 5.81843322574321e-115 < b

    1. Initial program 51.3

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

    2. Taylor expanded around inf 11.3

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))