Average Error: 33.1 → 10.7
Time: 27.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7503548021140933 \cdot 10^{-65}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -9.861592941135515 \cdot 10^{-102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{elif}\;b \le -4.884190020998732 \cdot 10^{-159}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 7.377921431051488 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.7503548021140933 \cdot 10^{-65}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -9.861592941135515 \cdot 10^{-102}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

\mathbf{elif}\;b \le -4.884190020998732 \cdot 10^{-159}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 7.377921431051488 \cdot 10^{+75}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r2086804 = b;
        double r2086805 = -r2086804;
        double r2086806 = r2086804 * r2086804;
        double r2086807 = 4.0;
        double r2086808 = a;
        double r2086809 = c;
        double r2086810 = r2086808 * r2086809;
        double r2086811 = r2086807 * r2086810;
        double r2086812 = r2086806 - r2086811;
        double r2086813 = sqrt(r2086812);
        double r2086814 = r2086805 - r2086813;
        double r2086815 = 2.0;
        double r2086816 = r2086815 * r2086808;
        double r2086817 = r2086814 / r2086816;
        return r2086817;
}
↓
double f(double a, double b, double c) {
        double r2086818 = b;
        double r2086819 = -2.7503548021140933e-65;
        bool r2086820 = r2086818 <= r2086819;
        double r2086821 = c;
        double r2086822 = r2086821 / r2086818;
        double r2086823 = -r2086822;
        double r2086824 = -9.861592941135515e-102;
        bool r2086825 = r2086818 <= r2086824;
        double r2086826 = -r2086818;
        double r2086827 = r2086818 * r2086818;
        double r2086828 = a;
        double r2086829 = r2086821 * r2086828;
        double r2086830 = 4.0;
        double r2086831 = r2086829 * r2086830;
        double r2086832 = r2086827 - r2086831;
        double r2086833 = sqrt(r2086832);
        double r2086834 = r2086826 - r2086833;
        double r2086835 = 2.0;
        double r2086836 = r2086828 * r2086835;
        double r2086837 = r2086834 / r2086836;
        double r2086838 = -4.884190020998732e-159;
        bool r2086839 = r2086818 <= r2086838;
        double r2086840 = 7.377921431051488e+75;
        bool r2086841 = r2086818 <= r2086840;
        double r2086842 = r2086818 / r2086828;
        double r2086843 = r2086822 - r2086842;
        double r2086844 = r2086841 ? r2086837 : r2086843;
        double r2086845 = r2086839 ? r2086823 : r2086844;
        double r2086846 = r2086825 ? r2086837 : r2086845;
        double r2086847 = r2086820 ? r2086823 : r2086846;
        return r2086847;
}

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

Original33.1
Target20.8
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.7503548021140933e-65 or -9.861592941135515e-102 < b < -4.884190020998732e-159

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 11.9

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

    3. Simplified11.9

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

    if -2.7503548021140933e-65 < b < -9.861592941135515e-102 or -4.884190020998732e-159 < b < 7.377921431051488e+75

    1. Initial program 12.4

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

    2. Taylor expanded around 0 12.4

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

    3. Simplified12.4

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

    if 7.377921431051488e+75 < b

    1. Initial program 39.3

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

    2. Taylor expanded around 0 39.3

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

    3. Simplified39.3

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

    4. Taylor expanded around inf 4.2

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7503548021140933 \cdot 10^{-65}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -9.861592941135515 \cdot 10^{-102}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{elif}\;b \le -4.884190020998732 \cdot 10^{-159}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 7.377921431051488 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))