Average Error: 20.3 → 6.6
Time: 17.5s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.328163156172201171875615953224398190771 \cdot 10^{-137}:\\ \;\;\;\;\left(b \cdot \frac{\frac{1}{z}}{c} + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.857809735758019969223840291451408280898 \cdot 10^{50}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.524854797087028066381557610408788493249 \cdot 10^{304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{t \cdot a}{c}\\ \end{array}\]
\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.328163156172201171875615953224398190771 \cdot 10^{-137}:\\
\;\;\;\;\left(b \cdot \frac{\frac{1}{z}}{c} + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\\

\mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.857809735758019969223840291451408280898 \cdot 10^{50}:\\
\;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}} - \left(a \cdot 4\right) \cdot t}{c}\\

\mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.524854797087028066381557610408788493249 \cdot 10^{304}:\\
\;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{t \cdot a}{c}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r610837 = x;
        double r610838 = 9.0;
        double r610839 = r610837 * r610838;
        double r610840 = y;
        double r610841 = r610839 * r610840;
        double r610842 = z;
        double r610843 = 4.0;
        double r610844 = r610842 * r610843;
        double r610845 = t;
        double r610846 = r610844 * r610845;
        double r610847 = a;
        double r610848 = r610846 * r610847;
        double r610849 = r610841 - r610848;
        double r610850 = b;
        double r610851 = r610849 + r610850;
        double r610852 = c;
        double r610853 = r610842 * r610852;
        double r610854 = r610851 / r610853;
        return r610854;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r610855 = x;
        double r610856 = 9.0;
        double r610857 = r610855 * r610856;
        double r610858 = y;
        double r610859 = r610857 * r610858;
        double r610860 = z;
        double r610861 = 4.0;
        double r610862 = r610860 * r610861;
        double r610863 = t;
        double r610864 = r610862 * r610863;
        double r610865 = a;
        double r610866 = r610864 * r610865;
        double r610867 = r610859 - r610866;
        double r610868 = b;
        double r610869 = r610867 + r610868;
        double r610870 = c;
        double r610871 = r610860 * r610870;
        double r610872 = r610869 / r610871;
        double r610873 = -2.328163156172201e-137;
        bool r610874 = r610872 <= r610873;
        double r610875 = 1.0;
        double r610876 = r610875 / r610860;
        double r610877 = r610876 / r610870;
        double r610878 = r610868 * r610877;
        double r610879 = r610855 * r610858;
        double r610880 = r610856 * r610879;
        double r610881 = r610875 / r610871;
        double r610882 = r610880 * r610881;
        double r610883 = r610878 + r610882;
        double r610884 = r610863 * r610865;
        double r610885 = r610884 / r610870;
        double r610886 = r610861 * r610885;
        double r610887 = r610883 - r610886;
        double r610888 = 6.85780973575802e+50;
        bool r610889 = r610872 <= r610888;
        double r610890 = r610859 + r610868;
        double r610891 = r610860 / r610890;
        double r610892 = r610875 / r610891;
        double r610893 = r610865 * r610861;
        double r610894 = r610893 * r610863;
        double r610895 = r610892 - r610894;
        double r610896 = r610895 / r610870;
        double r610897 = 6.524854797087028e+304;
        bool r610898 = r610872 <= r610897;
        double r610899 = r610868 / r610871;
        double r610900 = r610855 / r610860;
        double r610901 = r610856 * r610900;
        double r610902 = r610858 / r610870;
        double r610903 = r610901 * r610902;
        double r610904 = r610899 + r610903;
        double r610905 = r610904 - r610886;
        double r610906 = r610898 ? r610872 : r610905;
        double r610907 = r610889 ? r610896 : r610906;
        double r610908 = r610874 ? r610887 : r610907;
        return r610908;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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

Original20.3
Target14.1
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.328163156172201e-137

    1. Initial program 12.8

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified13.1

      \[\leadsto \color{blue}{\frac{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]

    3. Taylor expanded around 0 7.7

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]
    4. Using strategy rm

    5. Applied div-inv7.9

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \frac{1}{z \cdot c}\right)}\right) - 4 \cdot \frac{t \cdot a}{c}\]

    6. Applied associate-*r*8.0

      \[\leadsto \left(\frac{b}{z \cdot c} + \color{blue}{\left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}}\right) - 4 \cdot \frac{t \cdot a}{c}\]
    7. Using strategy rm

    8. Applied div-inv8.0

      \[\leadsto \left(\color{blue}{b \cdot \frac{1}{z \cdot c}} + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\]

    9. Simplified8.0

      \[\leadsto \left(b \cdot \color{blue}{\frac{1}{c \cdot z}} + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\]
    10. Using strategy rm

    11. Applied div-inv8.0

      \[\leadsto \left(b \cdot \color{blue}{\left(1 \cdot \frac{1}{c \cdot z}\right)} + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\]

    12. Simplified8.0

      \[\leadsto \left(b \cdot \left(1 \cdot \color{blue}{\frac{\frac{1}{z}}{c}}\right) + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\]

    if -2.328163156172201e-137 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 6.85780973575802e+50

    1. Initial program 14.7

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified1.2

      \[\leadsto \color{blue}{\frac{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
    3. Using strategy rm

    4. Applied clear-num1.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}}} - \left(a \cdot 4\right) \cdot t}{c}\]

    if 6.85780973575802e+50 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 6.524854797087028e+304

    1. Initial program 0.7

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    if 6.524854797087028e+304 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

    1. Initial program 63.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

    2. Simplified27.4

      \[\leadsto \color{blue}{\frac{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]

    3. Taylor expanded around 0 30.8

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]
    4. Using strategy rm

    5. Applied times-frac17.7

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\frac{x}{z} \cdot \frac{y}{c}\right)}\right) - 4 \cdot \frac{t \cdot a}{c}\]

    6. Applied associate-*r*17.7

      \[\leadsto \left(\frac{b}{z \cdot c} + \color{blue}{\left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}}\right) - 4 \cdot \frac{t \cdot a}{c}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.328163156172201171875615953224398190771 \cdot 10^{-137}:\\ \;\;\;\;\left(b \cdot \frac{\frac{1}{z}}{c} + \left(9 \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.857809735758019969223840291451408280898 \cdot 10^{50}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\left(x \cdot 9\right) \cdot y + b}} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.524854797087028066381557610408788493249 \cdot 10^{304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{t \cdot a}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041049e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))