Average Error: 34.0 → 10.6
Time: 4.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79775 = b;
        double r79776 = -r79775;
        double r79777 = r79775 * r79775;
        double r79778 = 4.0;
        double r79779 = a;
        double r79780 = c;
        double r79781 = r79779 * r79780;
        double r79782 = r79778 * r79781;
        double r79783 = r79777 - r79782;
        double r79784 = sqrt(r79783);
        double r79785 = r79776 - r79784;
        double r79786 = 2.0;
        double r79787 = r79786 * r79779;
        double r79788 = r79785 / r79787;
        return r79788;
}
↓
double f(double a, double b, double c) {
        double r79789 = b;
        double r79790 = -2.731633690849518e-121;
        bool r79791 = r79789 <= r79790;
        double r79792 = -1.0;
        double r79793 = c;
        double r79794 = r79793 / r79789;
        double r79795 = r79792 * r79794;
        double r79796 = 1.0273828621120979e+63;
        bool r79797 = r79789 <= r79796;
        double r79798 = 1.0;
        double r79799 = 2.0;
        double r79800 = a;
        double r79801 = r79799 * r79800;
        double r79802 = -r79789;
        double r79803 = r79789 * r79789;
        double r79804 = 4.0;
        double r79805 = r79800 * r79793;
        double r79806 = r79804 * r79805;
        double r79807 = r79803 - r79806;
        double r79808 = sqrt(r79807);
        double r79809 = r79802 - r79808;
        double r79810 = r79801 / r79809;
        double r79811 = r79798 / r79810;
        double r79812 = 1.0;
        double r79813 = r79789 / r79800;
        double r79814 = r79794 - r79813;
        double r79815 = r79812 * r79814;
        double r79816 = r79797 ? r79811 : r79815;
        double r79817 = r79791 ? r79795 : r79816;
        return r79817;
}

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
Target21.0
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.731633690849518e-121

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 11.5

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

    if -2.731633690849518e-121 < b < 1.0273828621120979e+63

    1. Initial program 12.1

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

    3. Applied clear-num12.2

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

    if 1.0273828621120979e+63 < b

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

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

    3. Simplified5.4

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))