Average Error: 33.2 → 10.7
Time: 19.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.2415082771065304 \cdot 10^{-131}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1555757 = b;
        double r1555758 = -r1555757;
        double r1555759 = r1555757 * r1555757;
        double r1555760 = 4.0;
        double r1555761 = a;
        double r1555762 = c;
        double r1555763 = r1555761 * r1555762;
        double r1555764 = r1555760 * r1555763;
        double r1555765 = r1555759 - r1555764;
        double r1555766 = sqrt(r1555765);
        double r1555767 = r1555758 - r1555766;
        double r1555768 = 2.0;
        double r1555769 = r1555768 * r1555761;
        double r1555770 = r1555767 / r1555769;
        return r1555770;
}
↓
double f(double a, double b, double c) {
        double r1555771 = b;
        double r1555772 = -2.2415082771065304e-131;
        bool r1555773 = r1555771 <= r1555772;
        double r1555774 = c;
        double r1555775 = r1555774 / r1555771;
        double r1555776 = -r1555775;
        double r1555777 = 2.559678284282607e+69;
        bool r1555778 = r1555771 <= r1555777;
        double r1555779 = -r1555771;
        double r1555780 = -4.0;
        double r1555781 = a;
        double r1555782 = r1555780 * r1555781;
        double r1555783 = r1555782 * r1555774;
        double r1555784 = r1555771 * r1555771;
        double r1555785 = r1555783 + r1555784;
        double r1555786 = sqrt(r1555785);
        double r1555787 = r1555779 - r1555786;
        double r1555788 = 2.0;
        double r1555789 = r1555781 * r1555788;
        double r1555790 = r1555787 / r1555789;
        double r1555791 = r1555771 / r1555781;
        double r1555792 = r1555775 - r1555791;
        double r1555793 = r1555778 ? r1555790 : r1555792;
        double r1555794 = r1555773 ? r1555776 : r1555793;
        return r1555794;
}

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

Original33.2
Target19.9
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 < -2.2415082771065304e-131

    1. Initial program 49.6

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

    2. Taylor expanded around -inf 12.4

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

    3. Simplified12.4

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

    if -2.2415082771065304e-131 < b < 2.559678284282607e+69

    1. Initial program 11.4

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

    2. Taylor expanded around inf 11.4

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

    3. Simplified11.4

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

    if 2.559678284282607e+69 < b

    1. Initial program 38.9

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))