Average Error: 34.2 → 15.1
Time: 11.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3358786167585806 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\ \mathbf{elif}\;b \le 1.94263717460376656 \cdot 10^{24}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{2}}{a}\\ \end{array}\]
\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 -1.3358786167585806 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\

\mathbf{elif}\;b \le 1.94263717460376656 \cdot 10^{24}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r155345 = b;
        double r155346 = -r155345;
        double r155347 = r155345 * r155345;
        double r155348 = 4.0;
        double r155349 = a;
        double r155350 = r155348 * r155349;
        double r155351 = c;
        double r155352 = r155350 * r155351;
        double r155353 = r155347 - r155352;
        double r155354 = sqrt(r155353);
        double r155355 = r155346 + r155354;
        double r155356 = 2.0;
        double r155357 = r155356 * r155349;
        double r155358 = r155355 / r155357;
        return r155358;
}
↓
double f(double a, double b, double c) {
        double r155359 = b;
        double r155360 = -1.3358786167585806e+154;
        bool r155361 = r155359 <= r155360;
        double r155362 = 2.0;
        double r155363 = a;
        double r155364 = c;
        double r155365 = r155363 * r155364;
        double r155366 = r155365 / r155359;
        double r155367 = r155362 * r155366;
        double r155368 = 2.0;
        double r155369 = r155368 * r155359;
        double r155370 = r155367 - r155369;
        double r155371 = r155370 / r155362;
        double r155372 = r155371 / r155363;
        double r155373 = 1.9426371746037666e+24;
        bool r155374 = r155359 <= r155373;
        double r155375 = r155359 * r155359;
        double r155376 = 4.0;
        double r155377 = r155376 * r155363;
        double r155378 = r155377 * r155364;
        double r155379 = r155375 - r155378;
        double r155380 = sqrt(r155379);
        double r155381 = r155380 - r155359;
        double r155382 = r155381 / r155362;
        double r155383 = r155382 / r155363;
        double r155384 = -2.0;
        double r155385 = r155384 * r155366;
        double r155386 = r155385 / r155362;
        double r155387 = r155386 / r155363;
        double r155388 = r155374 ? r155383 : r155387;
        double r155389 = r155361 ? r155372 : r155388;
        return r155389;
}

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

Original34.2
Target21.2
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3358786167585806e+154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 9.8

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

    if -1.3358786167585806e+154 < b < 1.9426371746037666e+24

    1. Initial program 16.1

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

    2. Simplified16.1

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

    if 1.9426371746037666e+24 < b

    1. Initial program 56.4

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

    2. Simplified56.4

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

    3. Taylor expanded around inf 15.2

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3358786167585806 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\ \mathbf{elif}\;b \le 1.94263717460376656 \cdot 10^{24}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))