Average Error: 34.6 → 6.9
Time: 18.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7604193036648139441831936:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -2.120900881031131292062715264701944285734 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)} - b}}\\ \mathbf{elif}\;b \le 2.345370025086597272923559832061889684617 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{4 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}} \cdot \frac{a}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -7604193036648139441831936:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le -2.120900881031131292062715264701944285734 \cdot 10^{-243}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)} - b}}\\

\mathbf{elif}\;b \le 2.345370025086597272923559832061889684617 \cdot 10^{84}:\\
\;\;\;\;\frac{1}{2} \cdot \left(\frac{4 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}} \cdot \frac{a}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r53757 = b;
        double r53758 = -r53757;
        double r53759 = r53757 * r53757;
        double r53760 = 4.0;
        double r53761 = a;
        double r53762 = c;
        double r53763 = r53761 * r53762;
        double r53764 = r53760 * r53763;
        double r53765 = r53759 - r53764;
        double r53766 = sqrt(r53765);
        double r53767 = r53758 + r53766;
        double r53768 = 2.0;
        double r53769 = r53768 * r53761;
        double r53770 = r53767 / r53769;
        return r53770;
}
↓
double f(double a, double b, double c) {
        double r53771 = b;
        double r53772 = -7.604193036648139e+24;
        bool r53773 = r53771 <= r53772;
        double r53774 = c;
        double r53775 = r53774 / r53771;
        double r53776 = a;
        double r53777 = r53771 / r53776;
        double r53778 = r53775 - r53777;
        double r53779 = 1.0;
        double r53780 = r53778 * r53779;
        double r53781 = -2.1209008810311313e-243;
        bool r53782 = r53771 <= r53781;
        double r53783 = 1.0;
        double r53784 = 2.0;
        double r53785 = r53776 * r53784;
        double r53786 = 4.0;
        double r53787 = r53786 * r53774;
        double r53788 = -r53787;
        double r53789 = 2.0;
        double r53790 = pow(r53771, r53789);
        double r53791 = fma(r53776, r53788, r53790);
        double r53792 = sqrt(r53791);
        double r53793 = r53792 - r53771;
        double r53794 = r53785 / r53793;
        double r53795 = r53783 / r53794;
        double r53796 = 2.3453700250865973e+84;
        bool r53797 = r53771 <= r53796;
        double r53798 = r53783 / r53784;
        double r53799 = -r53771;
        double r53800 = r53771 * r53771;
        double r53801 = fma(r53776, r53788, r53800);
        double r53802 = sqrt(r53801);
        double r53803 = r53799 - r53802;
        double r53804 = r53787 / r53803;
        double r53805 = r53776 / r53776;
        double r53806 = r53804 * r53805;
        double r53807 = r53798 * r53806;
        double r53808 = -1.0;
        double r53809 = r53775 * r53808;
        double r53810 = r53797 ? r53807 : r53809;
        double r53811 = r53782 ? r53795 : r53810;
        double r53812 = r53773 ? r53780 : r53811;
        return r53812;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.6
Target21.0
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.604193036648139e+24

    1. Initial program 35.7

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified6.3

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

    if -7.604193036648139e+24 < b < -2.1209008810311313e-243

    1. Initial program 9.4

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

    3. Applied clear-num9.5

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

    4. Simplified9.6

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

    if -2.1209008810311313e-243 < b < 2.3453700250865973e+84

    1. Initial program 29.5

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

    3. Applied flip-+29.6

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

    4. Simplified15.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot \left(c \cdot 4\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified15.9

      \[\leadsto \frac{\frac{0 + a \cdot \left(c \cdot 4\right)}{\color{blue}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot \left(-4\right), {b}^{2}\right)}}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity15.9

      \[\leadsto \frac{\frac{0 + a \cdot \left(c \cdot 4\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot \left(-4\right), {b}^{2}\right)}\right)}}}{2 \cdot a}\]

    8. Applied *-un-lft-identity15.9

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + a \cdot \left(c \cdot 4\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot \left(-4\right), {b}^{2}\right)}\right)}}{2 \cdot a}\]

    9. Applied times-frac15.9

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot \left(-4\right), {b}^{2}\right)}}}}{2 \cdot a}\]

    10. Applied times-frac15.9

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot \left(-4\right), {b}^{2}\right)}}}{a}}\]

    11. Simplified15.9

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot \left(-4\right), {b}^{2}\right)}}}{a}\]

    12. Simplified9.3

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

    if 2.3453700250865973e+84 < b

    1. Initial program 59.1

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

    2. Taylor expanded around inf 2.5

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

    3. Simplified2.5

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7604193036648139441831936:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -2.120900881031131292062715264701944285734 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, {b}^{2}\right)} - b}}\\ \mathbf{elif}\;b \le 2.345370025086597272923559832061889684617 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{4 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}} \cdot \frac{a}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))