Average Error: 34.2 → 10.3
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.429521559957367003973909183894614803551 \cdot 10^{-36}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.895438119410103188352046975315827374151 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)\\ \end{array}\]
\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 -3.429521559957367003973909183894614803551 \cdot 10^{-36}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 7.895438119410103188352046975315827374151 \cdot 10^{91}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r94047 = b;
        double r94048 = -r94047;
        double r94049 = r94047 * r94047;
        double r94050 = 4.0;
        double r94051 = a;
        double r94052 = c;
        double r94053 = r94051 * r94052;
        double r94054 = r94050 * r94053;
        double r94055 = r94049 - r94054;
        double r94056 = sqrt(r94055);
        double r94057 = r94048 - r94056;
        double r94058 = 2.0;
        double r94059 = r94058 * r94051;
        double r94060 = r94057 / r94059;
        return r94060;
}
↓
double f(double a, double b, double c) {
        double r94061 = b;
        double r94062 = -3.429521559957367e-36;
        bool r94063 = r94061 <= r94062;
        double r94064 = -1.0;
        double r94065 = c;
        double r94066 = r94065 / r94061;
        double r94067 = r94064 * r94066;
        double r94068 = 7.895438119410103e+91;
        bool r94069 = r94061 <= r94068;
        double r94070 = -r94061;
        double r94071 = 2.0;
        double r94072 = a;
        double r94073 = r94071 * r94072;
        double r94074 = r94070 / r94073;
        double r94075 = r94061 * r94061;
        double r94076 = 4.0;
        double r94077 = r94072 * r94065;
        double r94078 = r94076 * r94077;
        double r94079 = r94075 - r94078;
        double r94080 = sqrt(r94079);
        double r94081 = r94080 / r94073;
        double r94082 = r94074 - r94081;
        double r94083 = 0.5;
        double r94084 = r94061 / r94072;
        double r94085 = r94083 * r94084;
        double r94086 = 1.0;
        double r94087 = r94086 * r94066;
        double r94088 = r94085 - r94087;
        double r94089 = r94074 - r94088;
        double r94090 = r94069 ? r94082 : r94089;
        double r94091 = r94063 ? r94067 : r94090;
        return r94091;
}

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
Target21.1
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 < -3.429521559957367e-36

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 7.5

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

    if -3.429521559957367e-36 < b < 7.895438119410103e+91

    1. Initial program 14.7

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

    3. Applied div-sub14.7

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

    if 7.895438119410103e+91 < b

    1. Initial program 45.8

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

    3. Applied div-sub45.8

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

    4. Taylor expanded around inf 4.2

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{\left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.429521559957367003973909183894614803551 \cdot 10^{-36}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.895438119410103188352046975315827374151 \cdot 10^{91}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \left(0.5 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))