Average Error: 33.7 → 10.5
Time: 4.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3295118613703302 \cdot 10^{-13}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.29545095081340793 \cdot 10^{65}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.3295118613703302 \cdot 10^{-13}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.29545095081340793 \cdot 10^{65}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r88327 = b;
        double r88328 = -r88327;
        double r88329 = r88327 * r88327;
        double r88330 = 4.0;
        double r88331 = a;
        double r88332 = c;
        double r88333 = r88331 * r88332;
        double r88334 = r88330 * r88333;
        double r88335 = r88329 - r88334;
        double r88336 = sqrt(r88335);
        double r88337 = r88328 - r88336;
        double r88338 = 2.0;
        double r88339 = r88338 * r88331;
        double r88340 = r88337 / r88339;
        return r88340;
}
↓
double f(double a, double b, double c) {
        double r88341 = b;
        double r88342 = -1.3295118613703302e-13;
        bool r88343 = r88341 <= r88342;
        double r88344 = -1.0;
        double r88345 = c;
        double r88346 = r88345 / r88341;
        double r88347 = r88344 * r88346;
        double r88348 = 3.295450950813408e+65;
        bool r88349 = r88341 <= r88348;
        double r88350 = -r88341;
        double r88351 = r88341 * r88341;
        double r88352 = 4.0;
        double r88353 = a;
        double r88354 = r88353 * r88345;
        double r88355 = r88352 * r88354;
        double r88356 = r88351 - r88355;
        double r88357 = sqrt(r88356);
        double r88358 = r88350 - r88357;
        double r88359 = 1.0;
        double r88360 = 2.0;
        double r88361 = r88360 * r88353;
        double r88362 = r88359 / r88361;
        double r88363 = r88358 * r88362;
        double r88364 = 1.0;
        double r88365 = r88341 / r88353;
        double r88366 = r88346 - r88365;
        double r88367 = r88364 * r88366;
        double r88368 = r88349 ? r88363 : r88367;
        double r88369 = r88343 ? r88347 : r88368;
        return r88369;
}

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

Original33.7
Target21.0
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3295118613703302e-13

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 6.9

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

    if -1.3295118613703302e-13 < b < 3.295450950813408e+65

    1. Initial program 15.4

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

    3. Applied div-inv15.5

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

    if 3.295450950813408e+65 < b

    1. Initial program 40.0

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

    2. Taylor expanded around inf 4.7

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

    3. Simplified4.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3295118613703302 \cdot 10^{-13}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.29545095081340793 \cdot 10^{65}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))