Average Error: 33.8 → 9.9
Time: 4.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.823527231207463414371193958705943379769 \cdot 10^{100}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.823527231207463414371193958705943379769 \cdot 10^{100}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r75025 = b;
        double r75026 = -r75025;
        double r75027 = r75025 * r75025;
        double r75028 = 4.0;
        double r75029 = a;
        double r75030 = c;
        double r75031 = r75029 * r75030;
        double r75032 = r75028 * r75031;
        double r75033 = r75027 - r75032;
        double r75034 = sqrt(r75033);
        double r75035 = r75026 - r75034;
        double r75036 = 2.0;
        double r75037 = r75036 * r75029;
        double r75038 = r75035 / r75037;
        return r75038;
}
↓
double f(double a, double b, double c) {
        double r75039 = b;
        double r75040 = -1.3696943711263392e-83;
        bool r75041 = r75039 <= r75040;
        double r75042 = -1.0;
        double r75043 = c;
        double r75044 = r75043 / r75039;
        double r75045 = r75042 * r75044;
        double r75046 = 6.8235272312074634e+100;
        bool r75047 = r75039 <= r75046;
        double r75048 = -r75039;
        double r75049 = r75039 * r75039;
        double r75050 = 4.0;
        double r75051 = a;
        double r75052 = r75051 * r75043;
        double r75053 = r75050 * r75052;
        double r75054 = r75049 - r75053;
        double r75055 = sqrt(r75054);
        double r75056 = r75048 - r75055;
        double r75057 = 2.0;
        double r75058 = r75056 / r75057;
        double r75059 = 1.0;
        double r75060 = r75059 / r75051;
        double r75061 = r75058 * r75060;
        double r75062 = 1.0;
        double r75063 = r75039 / r75051;
        double r75064 = r75044 - r75063;
        double r75065 = r75062 * r75064;
        double r75066 = r75047 ? r75061 : r75065;
        double r75067 = r75041 ? r75045 : r75066;
        return r75067;
}

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

Original33.8
Target20.8
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3696943711263392e-83

    1. Initial program 53.2

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

    2. Taylor expanded around -inf 9.4

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

    if -1.3696943711263392e-83 < b < 6.8235272312074634e+100

    1. Initial program 12.3

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

    3. Applied div-inv12.5

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

    5. Applied *-un-lft-identity12.5

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

    6. Applied times-frac12.5

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

    7. Applied associate-*r*12.5

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

    8. Simplified12.5

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

    if 6.8235272312074634e+100 < b

    1. Initial program 46.4

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

    2. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.823527231207463414371193958705943379769 \cdot 10^{100}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))