Average Error: 20.5 → 9.5
Time: 5.6s
Precision: 64
\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.17579813555347906 \cdot 10^{-211}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z \cdot c}}{\frac{1}{y}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;t \le -1.25362070674570982 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\ \mathbf{elif}\;t \le 2.7046851978488097 \cdot 10^{-104}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 2.41220335967200437 \cdot 10^{70}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{x}{\frac{c}{\sqrt[3]{y}}}\right)\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \end{array}\]
\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}\;t \le -3.17579813555347906 \cdot 10^{-211}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z \cdot c}}{\frac{1}{y}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\

\mathbf{elif}\;t \le -1.25362070674570982 \cdot 10^{-303}:\\
\;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\

\mathbf{elif}\;t \le 2.7046851978488097 \cdot 10^{-104}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\

\mathbf{elif}\;t \le 2.41220335967200437 \cdot 10^{70}:\\
\;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{x}{\frac{c}{\sqrt[3]{y}}}\right)\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r735907 = x;
        double r735908 = 9.0;
        double r735909 = r735907 * r735908;
        double r735910 = y;
        double r735911 = r735909 * r735910;
        double r735912 = z;
        double r735913 = 4.0;
        double r735914 = r735912 * r735913;
        double r735915 = t;
        double r735916 = r735914 * r735915;
        double r735917 = a;
        double r735918 = r735916 * r735917;
        double r735919 = r735911 - r735918;
        double r735920 = b;
        double r735921 = r735919 + r735920;
        double r735922 = c;
        double r735923 = r735912 * r735922;
        double r735924 = r735921 / r735923;
        return r735924;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r735925 = t;
        double r735926 = -3.175798135553479e-211;
        bool r735927 = r735925 <= r735926;
        double r735928 = b;
        double r735929 = z;
        double r735930 = c;
        double r735931 = r735929 * r735930;
        double r735932 = r735928 / r735931;
        double r735933 = 9.0;
        double r735934 = x;
        double r735935 = r735934 / r735931;
        double r735936 = 1.0;
        double r735937 = y;
        double r735938 = r735936 / r735937;
        double r735939 = r735935 / r735938;
        double r735940 = r735933 * r735939;
        double r735941 = r735932 + r735940;
        double r735942 = 4.0;
        double r735943 = a;
        double r735944 = r735930 / r735925;
        double r735945 = r735943 / r735944;
        double r735946 = r735942 * r735945;
        double r735947 = r735941 - r735946;
        double r735948 = -1.2536207067457098e-303;
        bool r735949 = r735925 <= r735948;
        double r735950 = r735936 / r735929;
        double r735951 = r735934 * r735933;
        double r735952 = r735951 * r735937;
        double r735953 = r735929 * r735942;
        double r735954 = r735953 * r735925;
        double r735955 = r735954 * r735943;
        double r735956 = r735952 - r735955;
        double r735957 = r735956 + r735928;
        double r735958 = r735957 / r735930;
        double r735959 = r735950 * r735958;
        double r735960 = 2.7046851978488097e-104;
        bool r735961 = r735925 <= r735960;
        double r735962 = r735934 * r735937;
        double r735963 = r735962 / r735931;
        double r735964 = r735933 * r735963;
        double r735965 = r735932 + r735964;
        double r735966 = r735925 / r735930;
        double r735967 = r735943 * r735966;
        double r735968 = r735942 * r735967;
        double r735969 = r735965 - r735968;
        double r735970 = 2.4122033596720044e+70;
        bool r735971 = r735925 <= r735970;
        double r735972 = cbrt(r735937);
        double r735973 = r735972 * r735972;
        double r735974 = r735929 / r735973;
        double r735975 = r735936 / r735974;
        double r735976 = r735930 / r735972;
        double r735977 = r735934 / r735976;
        double r735978 = r735975 * r735977;
        double r735979 = r735933 * r735978;
        double r735980 = r735932 + r735979;
        double r735981 = r735980 - r735946;
        double r735982 = r735931 / r735937;
        double r735983 = r735934 / r735982;
        double r735984 = r735933 * r735983;
        double r735985 = r735932 + r735984;
        double r735986 = r735943 / r735930;
        double r735987 = r735986 * r735925;
        double r735988 = r735942 * r735987;
        double r735989 = r735985 - r735988;
        double r735990 = r735971 ? r735981 : r735989;
        double r735991 = r735961 ? r735969 : r735990;
        double r735992 = r735949 ? r735959 : r735991;
        double r735993 = r735927 ? r735947 : r735992;
        return r735993;
}

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

Original20.5
Target14.9
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if t < -3.175798135553479e-211

    1. Initial program 21.9

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

    2. Taylor expanded around 0 12.5

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

    4. Applied associate-/l*10.7

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

    6. Applied associate-/l*8.9

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

    8. Applied div-inv8.9

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

    9. Applied associate-/r*9.2

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

    if -3.175798135553479e-211 < t < -1.2536207067457098e-303

    1. Initial program 12.4

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

    3. Applied *-un-lft-identity12.4

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

    4. Applied times-frac11.1

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

    if -1.2536207067457098e-303 < t < 2.7046851978488097e-104

    1. Initial program 11.8

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

    2. Taylor expanded around 0 8.8

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

    4. Applied *-un-lft-identity8.8

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

    5. Applied times-frac10.6

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

    6. Simplified10.6

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

    if 2.7046851978488097e-104 < t < 2.4122033596720044e+70

    1. Initial program 19.1

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

    2. Taylor expanded around 0 10.6

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

    4. Applied associate-/l*10.6

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

    6. Applied associate-/l*8.9

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

    8. Applied add-cube-cbrt9.1

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

    9. Applied times-frac8.5

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

    10. Applied *-un-lft-identity8.5

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{\color{blue}{1 \cdot x}}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{c}{\sqrt[3]{y}}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\]

    11. Applied times-frac9.7

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\frac{1}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{x}{\frac{c}{\sqrt[3]{y}}}\right)}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\]

    if 2.4122033596720044e+70 < t

    1. Initial program 33.0

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

    2. Taylor expanded around 0 17.8

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

    4. Applied associate-/l*11.4

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

    6. Applied associate-/l*9.3

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

    8. Applied associate-/r/8.0

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \color{blue}{\left(\frac{a}{c} \cdot t\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.17579813555347906 \cdot 10^{-211}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{\frac{x}{z \cdot c}}{\frac{1}{y}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;t \le -1.25362070674570982 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{c}\\ \mathbf{elif}\;t \le 2.7046851978488097 \cdot 10^{-104}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \le 2.41220335967200437 \cdot 10^{70}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{1}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{x}{\frac{c}{\sqrt[3]{y}}}\right)\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \left(\frac{a}{c} \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)) -1.1001567408041051e-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)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* 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)))