Average Error: 34.4 → 9.1
Time: 20.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.118633551268419604580976770096584661265 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.547489674828777234150499172138545356488 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.118633551268419604580976770096584661265 \cdot 10^{-302}:\\
\;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}{a \cdot 2}\\

\mathbf{elif}\;b \le 4.547489674828777234150499172138545356488 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r4275365 = b;
        double r4275366 = -r4275365;
        double r4275367 = r4275365 * r4275365;
        double r4275368 = 4.0;
        double r4275369 = a;
        double r4275370 = c;
        double r4275371 = r4275369 * r4275370;
        double r4275372 = r4275368 * r4275371;
        double r4275373 = r4275367 - r4275372;
        double r4275374 = sqrt(r4275373);
        double r4275375 = r4275366 - r4275374;
        double r4275376 = 2.0;
        double r4275377 = r4275376 * r4275369;
        double r4275378 = r4275375 / r4275377;
        return r4275378;
}
↓
double f(double a, double b, double c) {
        double r4275379 = b;
        double r4275380 = -7.741777288939024e+81;
        bool r4275381 = r4275379 <= r4275380;
        double r4275382 = -1.0;
        double r4275383 = c;
        double r4275384 = r4275383 / r4275379;
        double r4275385 = r4275382 * r4275384;
        double r4275386 = -2.1186335512684196e-302;
        bool r4275387 = r4275379 <= r4275386;
        double r4275388 = a;
        double r4275389 = r4275388 * r4275383;
        double r4275390 = 4.0;
        double r4275391 = r4275389 * r4275390;
        double r4275392 = r4275379 * r4275379;
        double r4275393 = r4275392 - r4275391;
        double r4275394 = sqrt(r4275393);
        double r4275395 = r4275394 - r4275379;
        double r4275396 = r4275391 / r4275395;
        double r4275397 = 2.0;
        double r4275398 = r4275388 * r4275397;
        double r4275399 = r4275396 / r4275398;
        double r4275400 = 4.547489674828777e+101;
        bool r4275401 = r4275379 <= r4275400;
        double r4275402 = 1.0;
        double r4275403 = r4275402 / r4275398;
        double r4275404 = -r4275379;
        double r4275405 = r4275404 - r4275394;
        double r4275406 = r4275403 * r4275405;
        double r4275407 = r4275379 / r4275388;
        double r4275408 = r4275384 - r4275407;
        double r4275409 = 1.0;
        double r4275410 = r4275408 * r4275409;
        double r4275411 = r4275401 ? r4275406 : r4275410;
        double r4275412 = r4275387 ? r4275399 : r4275411;
        double r4275413 = r4275381 ? r4275385 : r4275412;
        return r4275413;
}

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

Original34.4
Target20.9
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.741777288939024e+81

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 2.9

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

    if -7.741777288939024e+81 < b < -2.1186335512684196e-302

    1. Initial program 31.7

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

    3. Applied flip--31.7

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

    4. Simplified16.5

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified16.5

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]

    if -2.1186335512684196e-302 < b < 4.547489674828777e+101

    1. Initial program 9.4

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

    3. Applied div-inv9.6

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

    if 4.547489674828777e+101 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

      \[\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.118633551268419604580976770096584661265 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.547489674828777234150499172138545356488 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))