Average Error: 34.3 → 10.3
Time: 5.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.51740022507215 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -2.125553485370055 \cdot 10^{-113}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.51740022507215 \cdot 10^{112}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r98388 = b;
        double r98389 = -r98388;
        double r98390 = r98388 * r98388;
        double r98391 = 4.0;
        double r98392 = a;
        double r98393 = c;
        double r98394 = r98392 * r98393;
        double r98395 = r98391 * r98394;
        double r98396 = r98390 - r98395;
        double r98397 = sqrt(r98396);
        double r98398 = r98389 - r98397;
        double r98399 = 2.0;
        double r98400 = r98399 * r98392;
        double r98401 = r98398 / r98400;
        return r98401;
}
↓
double f(double a, double b, double c) {
        double r98402 = b;
        double r98403 = -2.125553485370055e-113;
        bool r98404 = r98402 <= r98403;
        double r98405 = -1.0;
        double r98406 = c;
        double r98407 = r98406 / r98402;
        double r98408 = r98405 * r98407;
        double r98409 = 6.51740022507215e+112;
        bool r98410 = r98402 <= r98409;
        double r98411 = 1.0;
        double r98412 = 2.0;
        double r98413 = a;
        double r98414 = r98412 * r98413;
        double r98415 = -r98402;
        double r98416 = r98402 * r98402;
        double r98417 = 4.0;
        double r98418 = r98413 * r98406;
        double r98419 = r98417 * r98418;
        double r98420 = r98416 - r98419;
        double r98421 = sqrt(r98420);
        double r98422 = r98415 - r98421;
        double r98423 = r98414 / r98422;
        double r98424 = r98411 / r98423;
        double r98425 = r98402 / r98413;
        double r98426 = r98405 * r98425;
        double r98427 = r98410 ? r98424 : r98426;
        double r98428 = r98404 ? r98408 : r98427;
        return r98428;
}

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

Original34.3
Target21.2
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.125553485370055e-113

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 10.8

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

    if -2.125553485370055e-113 < b < 6.51740022507215e+112

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    if 6.51740022507215e+112 < b

    1. Initial program 49.8

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

    3. Applied clear-num49.9

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

    4. Taylor expanded around 0 2.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.51740022507215 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))