Average Error: 34.7 → 10.3
Time: 5.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.6537017569063518 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.0479007947857462 \cdot 10^{99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.6537017569063518 \cdot 10^{-82}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.0479007947857462 \cdot 10^{99}:\\
\;\;\;\;\frac{\left(-b\right) - \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 r72728 = b;
        double r72729 = -r72728;
        double r72730 = r72728 * r72728;
        double r72731 = 4.0;
        double r72732 = a;
        double r72733 = c;
        double r72734 = r72732 * r72733;
        double r72735 = r72731 * r72734;
        double r72736 = r72730 - r72735;
        double r72737 = sqrt(r72736);
        double r72738 = r72729 - r72737;
        double r72739 = 2.0;
        double r72740 = r72739 * r72732;
        double r72741 = r72738 / r72740;
        return r72741;
}
↓
double f(double a, double b, double c) {
        double r72742 = b;
        double r72743 = -4.653701756906352e-82;
        bool r72744 = r72742 <= r72743;
        double r72745 = -1.0;
        double r72746 = c;
        double r72747 = r72746 / r72742;
        double r72748 = r72745 * r72747;
        double r72749 = 1.0479007947857462e+99;
        bool r72750 = r72742 <= r72749;
        double r72751 = -r72742;
        double r72752 = r72742 * r72742;
        double r72753 = 4.0;
        double r72754 = a;
        double r72755 = r72754 * r72746;
        double r72756 = r72753 * r72755;
        double r72757 = r72752 - r72756;
        double r72758 = sqrt(r72757);
        double r72759 = r72751 - r72758;
        double r72760 = 2.0;
        double r72761 = r72760 * r72754;
        double r72762 = r72759 / r72761;
        double r72763 = 1.0;
        double r72764 = r72742 / r72754;
        double r72765 = r72747 - r72764;
        double r72766 = r72763 * r72765;
        double r72767 = r72750 ? r72762 : r72766;
        double r72768 = r72744 ? r72748 : r72767;
        return r72768;
}

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

Original34.7
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.653701756906352e-82

    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 9.2

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

    if -4.653701756906352e-82 < b < 1.0479007947857462e+99

    1. Initial program 13.5

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

    if 1.0479007947857462e+99 < b

    1. Initial program 47.7

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.6537017569063518 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.0479007947857462 \cdot 10^{99}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))