Average Error: 34.2 → 10.0
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r97803 = b;
        double r97804 = -r97803;
        double r97805 = r97803 * r97803;
        double r97806 = 4.0;
        double r97807 = a;
        double r97808 = c;
        double r97809 = r97807 * r97808;
        double r97810 = r97806 * r97809;
        double r97811 = r97805 - r97810;
        double r97812 = sqrt(r97811);
        double r97813 = r97804 - r97812;
        double r97814 = 2.0;
        double r97815 = r97814 * r97807;
        double r97816 = r97813 / r97815;
        return r97816;
}
↓
double f(double a, double b, double c) {
        double r97817 = b;
        double r97818 = -4.011579732710567e-81;
        bool r97819 = r97817 <= r97818;
        double r97820 = -1.0;
        double r97821 = c;
        double r97822 = r97821 / r97817;
        double r97823 = r97820 * r97822;
        double r97824 = 1.3176462918432122e+99;
        bool r97825 = r97817 <= r97824;
        double r97826 = -r97817;
        double r97827 = 2.0;
        double r97828 = a;
        double r97829 = r97827 * r97828;
        double r97830 = r97826 / r97829;
        double r97831 = r97817 * r97817;
        double r97832 = 4.0;
        double r97833 = r97828 * r97821;
        double r97834 = r97832 * r97833;
        double r97835 = r97831 - r97834;
        double r97836 = sqrt(r97835);
        double r97837 = r97836 / r97829;
        double r97838 = r97830 - r97837;
        double r97839 = 1.0;
        double r97840 = r97817 / r97828;
        double r97841 = r97822 - r97840;
        double r97842 = r97839 * r97841;
        double r97843 = r97825 ? r97838 : r97842;
        double r97844 = r97819 ? r97823 : r97843;
        return r97844;
}

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
Target21.5
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.011579732710567e-81

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.4

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

    if -4.011579732710567e-81 < b < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied div-sub12.9

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

    if 1.3176462918432122e+99 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))