Average Error: 33.6 → 10.2
Time: 13.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.260961702089070630848300788408824469286 \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 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r38536 = b;
        double r38537 = -r38536;
        double r38538 = r38536 * r38536;
        double r38539 = 4.0;
        double r38540 = a;
        double r38541 = c;
        double r38542 = r38540 * r38541;
        double r38543 = r38539 * r38542;
        double r38544 = r38538 - r38543;
        double r38545 = sqrt(r38544);
        double r38546 = r38537 + r38545;
        double r38547 = 2.0;
        double r38548 = r38547 * r38540;
        double r38549 = r38546 / r38548;
        return r38549;
}
↓
double f(double a, double b, double c) {
        double r38550 = b;
        double r38551 = -1.2609617020890706e+118;
        bool r38552 = r38550 <= r38551;
        double r38553 = 1.0;
        double r38554 = c;
        double r38555 = r38554 / r38550;
        double r38556 = a;
        double r38557 = r38550 / r38556;
        double r38558 = r38555 - r38557;
        double r38559 = r38553 * r38558;
        double r38560 = 5.81843322574321e-115;
        bool r38561 = r38550 <= r38560;
        double r38562 = -r38550;
        double r38563 = r38550 * r38550;
        double r38564 = 4.0;
        double r38565 = r38556 * r38554;
        double r38566 = r38564 * r38565;
        double r38567 = r38563 - r38566;
        double r38568 = sqrt(r38567);
        double r38569 = r38562 + r38568;
        double r38570 = 2.0;
        double r38571 = r38570 * r38556;
        double r38572 = r38569 / r38571;
        double r38573 = -1.0;
        double r38574 = r38573 * r38555;
        double r38575 = r38561 ? r38572 : r38574;
        double r38576 = r38552 ? r38559 : r38575;
        return r38576;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2609617020890706e+118

    1. Initial program 51.6

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

    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

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

    if 5.81843322574321e-115 < b

    1. Initial program 51.3

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

    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

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

Reproduce

herbie shell --seed 2019298 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))