Average Error: 34.4 → 7.0
Time: 9.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\

\mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r74807 = b;
        double r74808 = -r74807;
        double r74809 = r74807 * r74807;
        double r74810 = 4.0;
        double r74811 = a;
        double r74812 = c;
        double r74813 = r74811 * r74812;
        double r74814 = r74810 * r74813;
        double r74815 = r74809 - r74814;
        double r74816 = sqrt(r74815);
        double r74817 = r74808 - r74816;
        double r74818 = 2.0;
        double r74819 = r74818 * r74811;
        double r74820 = r74817 / r74819;
        return r74820;
}
↓
double f(double a, double b, double c) {
        double r74821 = b;
        double r74822 = -1.457738542065717e+153;
        bool r74823 = r74821 <= r74822;
        double r74824 = -1.0;
        double r74825 = c;
        double r74826 = r74825 / r74821;
        double r74827 = r74824 * r74826;
        double r74828 = -2.695863739873929e-295;
        bool r74829 = r74821 <= r74828;
        double r74830 = 2.0;
        double r74831 = r74830 * r74825;
        double r74832 = 4.0;
        double r74833 = a;
        double r74834 = r74833 * r74825;
        double r74835 = r74832 * r74834;
        double r74836 = -r74835;
        double r74837 = fma(r74821, r74821, r74836);
        double r74838 = sqrt(r74837);
        double r74839 = r74838 - r74821;
        double r74840 = r74831 / r74839;
        double r74841 = 1.1912031425131646e+117;
        bool r74842 = r74821 <= r74841;
        double r74843 = 1.0;
        double r74844 = r74830 * r74833;
        double r74845 = -r74821;
        double r74846 = r74821 * r74821;
        double r74847 = r74846 - r74835;
        double r74848 = sqrt(r74847);
        double r74849 = r74845 - r74848;
        double r74850 = r74844 / r74849;
        double r74851 = r74843 / r74850;
        double r74852 = r74821 / r74833;
        double r74853 = r74824 * r74852;
        double r74854 = r74842 ? r74851 : r74853;
        double r74855 = r74829 ? r74840 : r74854;
        double r74856 = r74823 ? r74827 : r74855;
        return r74856;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.2
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.457738542065717e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.3

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

    if -1.457738542065717e+153 < b < -2.695863739873929e-295

    1. Initial program 35.5

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

    3. Applied flip--35.5

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied div-inv16.6

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

    9. Applied associate-*l/15.1

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

    10. Simplified15.0

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

    11. Taylor expanded around 0 8.3

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

    if -2.695863739873929e-295 < b < 1.1912031425131646e+117

    1. Initial program 9.8

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

    3. Applied clear-num9.9

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

    if 1.1912031425131646e+117 < b

    1. Initial program 50.7

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

    3. Applied flip--63.7

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

    4. Simplified62.7

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

    5. Simplified62.7

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

    6. Taylor expanded around 0 4.0

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.457738542065716919858398723449020930628 \cdot 10^{153}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.695863739873928877277501764874065503226 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{elif}\;b \le 1.191203142513164639216663918436976252985 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))