Average Error: 34.5 → 9.9
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.52779163831840318 \cdot 10^{117}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r51455 = b;
        double r51456 = -r51455;
        double r51457 = r51455 * r51455;
        double r51458 = 4.0;
        double r51459 = a;
        double r51460 = r51458 * r51459;
        double r51461 = c;
        double r51462 = r51460 * r51461;
        double r51463 = r51457 - r51462;
        double r51464 = sqrt(r51463);
        double r51465 = r51456 + r51464;
        double r51466 = 2.0;
        double r51467 = r51466 * r51459;
        double r51468 = r51465 / r51467;
        return r51468;
}
↓
double f(double a, double b, double c) {
        double r51469 = b;
        double r51470 = -1.5277916383184032e+117;
        bool r51471 = r51469 <= r51470;
        double r51472 = 1.0;
        double r51473 = c;
        double r51474 = r51473 / r51469;
        double r51475 = a;
        double r51476 = r51469 / r51475;
        double r51477 = r51474 - r51476;
        double r51478 = r51472 * r51477;
        double r51479 = 4.3062534203630095e-45;
        bool r51480 = r51469 <= r51479;
        double r51481 = 1.0;
        double r51482 = 2.0;
        double r51483 = r51482 * r51475;
        double r51484 = -r51469;
        double r51485 = r51469 * r51469;
        double r51486 = 4.0;
        double r51487 = r51486 * r51475;
        double r51488 = r51487 * r51473;
        double r51489 = r51485 - r51488;
        double r51490 = sqrt(r51489);
        double r51491 = r51484 + r51490;
        double r51492 = r51483 / r51491;
        double r51493 = r51481 / r51492;
        double r51494 = -1.0;
        double r51495 = r51494 * r51474;
        double r51496 = r51480 ? r51493 : r51495;
        double r51497 = r51471 ? r51478 : r51496;
        return r51497;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5277916383184032e+117

    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 3.7

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

    3. Simplified3.7

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

    if -1.5277916383184032e+117 < b < 4.3062534203630095e-45

    1. Initial program 13.6

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

    3. Applied clear-num13.7

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

    if 4.3062534203630095e-45 < b

    1. Initial program 54.8

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.52779163831840318 \cdot 10^{117}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.3062534203630095 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))