Average Error: 34.0 → 9.6
Time: 5.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.5940112039867074 \cdot 10^{100}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\
\;\;\;\;\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{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r88043 = b;
        double r88044 = -r88043;
        double r88045 = r88043 * r88043;
        double r88046 = 4.0;
        double r88047 = a;
        double r88048 = c;
        double r88049 = r88047 * r88048;
        double r88050 = r88046 * r88049;
        double r88051 = r88045 - r88050;
        double r88052 = sqrt(r88051);
        double r88053 = r88044 + r88052;
        double r88054 = 2.0;
        double r88055 = r88054 * r88047;
        double r88056 = r88053 / r88055;
        return r88056;
}
↓
double f(double a, double b, double c) {
        double r88057 = b;
        double r88058 = -3.5940112039867074e+100;
        bool r88059 = r88057 <= r88058;
        double r88060 = 1.0;
        double r88061 = c;
        double r88062 = r88061 / r88057;
        double r88063 = a;
        double r88064 = r88057 / r88063;
        double r88065 = r88062 - r88064;
        double r88066 = r88060 * r88065;
        double r88067 = 2.267195199467958e-82;
        bool r88068 = r88057 <= r88067;
        double r88069 = 1.0;
        double r88070 = 2.0;
        double r88071 = r88070 * r88063;
        double r88072 = -r88057;
        double r88073 = r88057 * r88057;
        double r88074 = 4.0;
        double r88075 = r88063 * r88061;
        double r88076 = r88074 * r88075;
        double r88077 = r88073 - r88076;
        double r88078 = sqrt(r88077);
        double r88079 = r88072 + r88078;
        double r88080 = r88071 / r88079;
        double r88081 = r88069 / r88080;
        double r88082 = -1.0;
        double r88083 = r88082 * r88062;
        double r88084 = r88068 ? r88081 : r88083;
        double r88085 = r88059 ? r88066 : r88084;
        return r88085;
}

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

Original34.0
Target20.7
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.5940112039867074e+100

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 3.8

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

    3. Simplified3.8

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

    if -3.5940112039867074e+100 < b < 2.267195199467958e-82

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 2.267195199467958e-82 < b

    1. Initial program 52.9

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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