Average Error: 33.3 → 10.6
Time: 22.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.424937854855119 \cdot 10^{-129}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.912332224813067 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -8.424937854855119 \cdot 10^{-129}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 3.912332224813067 \cdot 10^{+23}:\\
\;\;\;\;\frac{1}{\frac{2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1444769 = b;
        double r1444770 = -r1444769;
        double r1444771 = r1444769 * r1444769;
        double r1444772 = 4.0;
        double r1444773 = a;
        double r1444774 = c;
        double r1444775 = r1444773 * r1444774;
        double r1444776 = r1444772 * r1444775;
        double r1444777 = r1444771 - r1444776;
        double r1444778 = sqrt(r1444777);
        double r1444779 = r1444770 - r1444778;
        double r1444780 = 2.0;
        double r1444781 = r1444780 * r1444773;
        double r1444782 = r1444779 / r1444781;
        return r1444782;
}
↓
double f(double a, double b, double c) {
        double r1444783 = b;
        double r1444784 = -8.424937854855119e-129;
        bool r1444785 = r1444783 <= r1444784;
        double r1444786 = c;
        double r1444787 = r1444786 / r1444783;
        double r1444788 = -r1444787;
        double r1444789 = 3.912332224813067e+23;
        bool r1444790 = r1444783 <= r1444789;
        double r1444791 = 1.0;
        double r1444792 = 2.0;
        double r1444793 = -r1444783;
        double r1444794 = -4.0;
        double r1444795 = a;
        double r1444796 = r1444795 * r1444786;
        double r1444797 = r1444794 * r1444796;
        double r1444798 = fma(r1444783, r1444783, r1444797);
        double r1444799 = sqrt(r1444798);
        double r1444800 = r1444793 - r1444799;
        double r1444801 = r1444792 / r1444800;
        double r1444802 = r1444801 * r1444795;
        double r1444803 = r1444791 / r1444802;
        double r1444804 = r1444783 / r1444795;
        double r1444805 = r1444787 - r1444804;
        double r1444806 = r1444790 ? r1444803 : r1444805;
        double r1444807 = r1444785 ? r1444788 : r1444806;
        return r1444807;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.424937854855119e-129

    1. Initial program 50.5

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

    2. Taylor expanded around -inf 11.4

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

    3. Simplified11.4

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

    if -8.424937854855119e-129 < b < 3.912332224813067e+23

    1. Initial program 12.3

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

    3. Applied clear-num12.5

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

    4. Simplified12.5

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

    if 3.912332224813067e+23 < b

    1. Initial program 32.6

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

    2. Taylor expanded around inf 6.2

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.424937854855119 \cdot 10^{-129}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.912332224813067 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{\frac{2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}} \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(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)))