Average Error: 34.5 → 6.3
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.010589257950129889712053784076648301115 \cdot 10^{115}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.051859552149432150298271086580779209751 \cdot 10^{-283}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{elif}\;b \le 1.274921840087524160396938471478240294554 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.010589257950129889712053784076648301115 \cdot 10^{115}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.051859552149432150298271086580779209751 \cdot 10^{-283}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

\mathbf{elif}\;b \le 1.274921840087524160396938471478240294554 \cdot 10^{104}:\\
\;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r86911 = b;
        double r86912 = -r86911;
        double r86913 = r86911 * r86911;
        double r86914 = 4.0;
        double r86915 = a;
        double r86916 = c;
        double r86917 = r86915 * r86916;
        double r86918 = r86914 * r86917;
        double r86919 = r86913 - r86918;
        double r86920 = sqrt(r86919);
        double r86921 = r86912 + r86920;
        double r86922 = 2.0;
        double r86923 = r86922 * r86915;
        double r86924 = r86921 / r86923;
        return r86924;
}
↓
double f(double a, double b, double c) {
        double r86925 = b;
        double r86926 = -3.01058925795013e+115;
        bool r86927 = r86925 <= r86926;
        double r86928 = 1.0;
        double r86929 = c;
        double r86930 = r86929 / r86925;
        double r86931 = a;
        double r86932 = r86925 / r86931;
        double r86933 = r86930 - r86932;
        double r86934 = r86928 * r86933;
        double r86935 = -1.0518595521494322e-283;
        bool r86936 = r86925 <= r86935;
        double r86937 = 1.0;
        double r86938 = 2.0;
        double r86939 = r86938 * r86931;
        double r86940 = -r86925;
        double r86941 = r86925 * r86925;
        double r86942 = 4.0;
        double r86943 = r86931 * r86929;
        double r86944 = r86942 * r86943;
        double r86945 = r86941 - r86944;
        double r86946 = sqrt(r86945);
        double r86947 = r86940 + r86946;
        double r86948 = r86939 / r86947;
        double r86949 = r86937 / r86948;
        double r86950 = 1.2749218400875242e+104;
        bool r86951 = r86925 <= r86950;
        double r86952 = 0.5;
        double r86953 = r86929 / r86952;
        double r86954 = r86940 - r86946;
        double r86955 = r86953 / r86954;
        double r86956 = -1.0;
        double r86957 = r86956 * r86930;
        double r86958 = r86951 ? r86955 : r86957;
        double r86959 = r86936 ? r86949 : r86958;
        double r86960 = r86927 ? r86934 : r86959;
        return r86960;
}

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
Target21.1
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.01058925795013e+115

    1. Initial program 50.4

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

    2. Taylor expanded around -inf 3.2

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

    3. Simplified3.2

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

    if -3.01058925795013e+115 < b < -1.0518595521494322e-283

    1. Initial program 8.8

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

    3. Applied clear-num9.0

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

    if -1.0518595521494322e-283 < b < 1.2749218400875242e+104

    1. Initial program 31.5

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

    3. Applied flip-+31.6

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

    4. Simplified15.4

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

    6. Applied clear-num15.5

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

    7. Simplified14.9

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

    8. Taylor expanded around 0 8.6

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]
    9. Using strategy rm

    10. Applied associate-/r*8.4

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{0.5}{c}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    11. Simplified8.3

      \[\leadsto \frac{\color{blue}{\frac{c}{0.5}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

    if 1.2749218400875242e+104 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.010589257950129889712053784076648301115 \cdot 10^{115}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.051859552149432150298271086580779209751 \cdot 10^{-283}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{elif}\;b \le 1.274921840087524160396938471478240294554 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))