Average Error: 34.3 → 6.5
Time: 15.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.453262666864975976658380182242705976906 \cdot 10^{76}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{2}{4}}{c}}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \le 1.180913557368329815252913260832369757763 \cdot 10^{84}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.453262666864975976658380182242705976906 \cdot 10^{76}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{2}{4}}{c}}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\

\mathbf{elif}\;b \le 1.180913557368329815252913260832369757763 \cdot 10^{84}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r64907 = b;
        double r64908 = -r64907;
        double r64909 = r64907 * r64907;
        double r64910 = 4.0;
        double r64911 = a;
        double r64912 = c;
        double r64913 = r64911 * r64912;
        double r64914 = r64910 * r64913;
        double r64915 = r64909 - r64914;
        double r64916 = sqrt(r64915);
        double r64917 = r64908 - r64916;
        double r64918 = 2.0;
        double r64919 = r64918 * r64911;
        double r64920 = r64917 / r64919;
        return r64920;
}
↓
double f(double a, double b, double c) {
        double r64921 = b;
        double r64922 = -2.453262666864976e+76;
        bool r64923 = r64921 <= r64922;
        double r64924 = -1.0;
        double r64925 = c;
        double r64926 = r64925 / r64921;
        double r64927 = r64924 * r64926;
        double r64928 = 8.958852798091287e-209;
        bool r64929 = r64921 <= r64928;
        double r64930 = 1.0;
        double r64931 = 2.0;
        double r64932 = 4.0;
        double r64933 = r64931 / r64932;
        double r64934 = r64933 / r64925;
        double r64935 = r64930 / r64934;
        double r64936 = r64921 * r64921;
        double r64937 = a;
        double r64938 = r64925 * r64937;
        double r64939 = r64932 * r64938;
        double r64940 = r64936 - r64939;
        double r64941 = sqrt(r64940);
        double r64942 = r64941 - r64921;
        double r64943 = r64935 / r64942;
        double r64944 = 1.1809135573683298e+84;
        bool r64945 = r64921 <= r64944;
        double r64946 = r64931 * r64937;
        double r64947 = r64921 / r64946;
        double r64948 = -r64947;
        double r64949 = r64925 * r64932;
        double r64950 = r64949 * r64937;
        double r64951 = r64936 - r64950;
        double r64952 = sqrt(r64951);
        double r64953 = r64952 / r64931;
        double r64954 = r64953 / r64937;
        double r64955 = r64948 - r64954;
        double r64956 = 1.0;
        double r64957 = r64921 / r64937;
        double r64958 = r64926 - r64957;
        double r64959 = r64956 * r64958;
        double r64960 = r64945 ? r64955 : r64959;
        double r64961 = r64929 ? r64943 : r64960;
        double r64962 = r64923 ? r64927 : r64961;
        return r64962;
}

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

Original34.3
Target21.1
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.453262666864976e+76

    1. Initial program 58.6

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

    2. Simplified58.6

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

    3. Taylor expanded around -inf 3.0

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

    if -2.453262666864976e+76 < b < 8.958852798091287e-209

    1. Initial program 29.1

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

    2. Simplified29.1

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

    4. Applied flip--29.2

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

    5. Simplified17.1

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

    6. Simplified17.1

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

    8. Applied *-un-lft-identity17.1

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

    9. Applied *-un-lft-identity17.1

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

    10. Applied times-frac17.1

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

    11. Applied associate-/l*17.2

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

    12. Simplified10.9

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

    14. Applied div-inv10.9

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

    15. Simplified10.6

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

    if 8.958852798091287e-209 < b < 1.1809135573683298e+84

    1. Initial program 7.0

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

    2. Simplified7.0

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

    4. Applied div-sub7.0

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

    5. Simplified7.0

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

    if 1.1809135573683298e+84 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 3.5

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

    4. Simplified3.5

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.453262666864975976658380182242705976906 \cdot 10^{76}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{2}{4}}{c}}}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}\\ \mathbf{elif}\;b \le 1.180913557368329815252913260832369757763 \cdot 10^{84}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\frac{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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