Average Error: 33.5 → 10.1
Time: 7.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.0323767944871679 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\
\;\;\;\;\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 r130370 = b;
        double r130371 = -r130370;
        double r130372 = r130370 * r130370;
        double r130373 = 4.0;
        double r130374 = a;
        double r130375 = r130373 * r130374;
        double r130376 = c;
        double r130377 = r130375 * r130376;
        double r130378 = r130372 - r130377;
        double r130379 = sqrt(r130378);
        double r130380 = r130371 + r130379;
        double r130381 = 2.0;
        double r130382 = r130381 * r130374;
        double r130383 = r130380 / r130382;
        return r130383;
}
↓
double f(double a, double b, double c) {
        double r130384 = b;
        double r130385 = -4.032376794487168e+127;
        bool r130386 = r130384 <= r130385;
        double r130387 = 1.0;
        double r130388 = c;
        double r130389 = r130388 / r130384;
        double r130390 = a;
        double r130391 = r130384 / r130390;
        double r130392 = r130389 - r130391;
        double r130393 = r130387 * r130392;
        double r130394 = 1.1752867948836086e-69;
        bool r130395 = r130384 <= r130394;
        double r130396 = 1.0;
        double r130397 = 2.0;
        double r130398 = r130397 * r130390;
        double r130399 = -r130384;
        double r130400 = r130384 * r130384;
        double r130401 = 4.0;
        double r130402 = r130401 * r130390;
        double r130403 = r130402 * r130388;
        double r130404 = r130400 - r130403;
        double r130405 = sqrt(r130404);
        double r130406 = r130399 + r130405;
        double r130407 = r130398 / r130406;
        double r130408 = r130396 / r130407;
        double r130409 = -1.0;
        double r130410 = r130409 * r130389;
        double r130411 = r130395 ? r130408 : r130410;
        double r130412 = r130386 ? r130393 : r130411;
        return r130412;
}

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

Original33.5
Target20.6
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.032376794487168e+127

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -4.032376794487168e+127 < b < 1.1752867948836086e-69

    1. Initial program 12.7

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

    3. Applied clear-num12.8

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

    if 1.1752867948836086e-69 < b

    1. Initial program 53.9

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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)))