Average Error: 34.5 → 10.2
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\
\;\;\;\;\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 r63902 = b;
        double r63903 = -r63902;
        double r63904 = r63902 * r63902;
        double r63905 = 4.0;
        double r63906 = a;
        double r63907 = c;
        double r63908 = r63906 * r63907;
        double r63909 = r63905 * r63908;
        double r63910 = r63904 - r63909;
        double r63911 = sqrt(r63910);
        double r63912 = r63903 - r63911;
        double r63913 = 2.0;
        double r63914 = r63913 * r63906;
        double r63915 = r63912 / r63914;
        return r63915;
}
↓
double f(double a, double b, double c) {
        double r63916 = b;
        double r63917 = -4.706781135059312e-92;
        bool r63918 = r63916 <= r63917;
        double r63919 = -1.0;
        double r63920 = c;
        double r63921 = r63920 / r63916;
        double r63922 = r63919 * r63921;
        double r63923 = 5.722235152988638e+98;
        bool r63924 = r63916 <= r63923;
        double r63925 = -r63916;
        double r63926 = r63916 * r63916;
        double r63927 = 4.0;
        double r63928 = a;
        double r63929 = r63928 * r63920;
        double r63930 = r63927 * r63929;
        double r63931 = r63926 - r63930;
        double r63932 = sqrt(r63931);
        double r63933 = r63925 - r63932;
        double r63934 = 2.0;
        double r63935 = r63934 * r63928;
        double r63936 = r63933 / r63935;
        double r63937 = 1.0;
        double r63938 = r63916 / r63928;
        double r63939 = r63921 - r63938;
        double r63940 = r63937 * r63939;
        double r63941 = r63924 ? r63936 : r63940;
        double r63942 = r63918 ? r63922 : r63941;
        return r63942;
}

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

Original34.5
Target21.5
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.706781135059312e-92

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 10.3

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

    if -4.706781135059312e-92 < b < 5.722235152988638e+98

    1. Initial program 12.7

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

    3. Applied div-inv12.8

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

    5. Applied un-div-inv12.7

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

    if 5.722235152988638e+98 < b

    1. Initial program 47.2

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

    2. Taylor expanded around inf 3.6

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

    3. Simplified3.6

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))