Average Error: 5.4 → 2.0
Time: 1.5m
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.7461996372679715 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(18.0 \cdot t\right) \cdot z\right) \cdot x\right) \cdot y - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;y \le 3.73999027982069 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right), t, \left(b \cdot c - \mathsf{fma}\left(\left(4.0 \cdot i\right), x, \left(k \cdot \left(j \cdot 27.0\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(x \cdot \left(t \cdot \left(z \cdot 18.0\right)\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.7461996372679715 \cdot 10^{-56}:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(18.0 \cdot t\right) \cdot z\right) \cdot x\right) \cdot y - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\

\mathbf{elif}\;y \le 3.73999027982069 \cdot 10^{-174}:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right), t, \left(b \cdot c - \mathsf{fma}\left(\left(4.0 \cdot i\right), x, \left(k \cdot \left(j \cdot 27.0\right)\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(y \cdot \left(x \cdot \left(t \cdot \left(z \cdot 18.0\right)\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r24198015 = x;
        double r24198016 = 18.0;
        double r24198017 = r24198015 * r24198016;
        double r24198018 = y;
        double r24198019 = r24198017 * r24198018;
        double r24198020 = z;
        double r24198021 = r24198019 * r24198020;
        double r24198022 = t;
        double r24198023 = r24198021 * r24198022;
        double r24198024 = a;
        double r24198025 = 4.0;
        double r24198026 = r24198024 * r24198025;
        double r24198027 = r24198026 * r24198022;
        double r24198028 = r24198023 - r24198027;
        double r24198029 = b;
        double r24198030 = c;
        double r24198031 = r24198029 * r24198030;
        double r24198032 = r24198028 + r24198031;
        double r24198033 = r24198015 * r24198025;
        double r24198034 = i;
        double r24198035 = r24198033 * r24198034;
        double r24198036 = r24198032 - r24198035;
        double r24198037 = j;
        double r24198038 = 27.0;
        double r24198039 = r24198037 * r24198038;
        double r24198040 = k;
        double r24198041 = r24198039 * r24198040;
        double r24198042 = r24198036 - r24198041;
        return r24198042;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r24198043 = y;
        double r24198044 = -1.7461996372679715e-56;
        bool r24198045 = r24198043 <= r24198044;
        double r24198046 = b;
        double r24198047 = c;
        double r24198048 = r24198046 * r24198047;
        double r24198049 = 18.0;
        double r24198050 = t;
        double r24198051 = r24198049 * r24198050;
        double r24198052 = z;
        double r24198053 = r24198051 * r24198052;
        double r24198054 = x;
        double r24198055 = r24198053 * r24198054;
        double r24198056 = r24198055 * r24198043;
        double r24198057 = a;
        double r24198058 = 4.0;
        double r24198059 = r24198057 * r24198058;
        double r24198060 = r24198059 * r24198050;
        double r24198061 = r24198056 - r24198060;
        double r24198062 = r24198048 + r24198061;
        double r24198063 = r24198058 * r24198054;
        double r24198064 = i;
        double r24198065 = r24198063 * r24198064;
        double r24198066 = r24198062 - r24198065;
        double r24198067 = k;
        double r24198068 = j;
        double r24198069 = 27.0;
        double r24198070 = r24198068 * r24198069;
        double r24198071 = r24198067 * r24198070;
        double r24198072 = r24198066 - r24198071;
        double r24198073 = 3.73999027982069e-174;
        bool r24198074 = r24198043 <= r24198073;
        double r24198075 = r24198052 * r24198043;
        double r24198076 = r24198054 * r24198049;
        double r24198077 = r24198075 * r24198076;
        double r24198078 = r24198077 - r24198059;
        double r24198079 = r24198058 * r24198064;
        double r24198080 = fma(r24198079, r24198054, r24198071);
        double r24198081 = r24198048 - r24198080;
        double r24198082 = fma(r24198078, r24198050, r24198081);
        double r24198083 = r24198052 * r24198049;
        double r24198084 = r24198050 * r24198083;
        double r24198085 = r24198054 * r24198084;
        double r24198086 = r24198043 * r24198085;
        double r24198087 = r24198086 - r24198060;
        double r24198088 = r24198087 + r24198048;
        double r24198089 = r24198088 - r24198065;
        double r24198090 = r24198068 * r24198067;
        double r24198091 = r24198090 * r24198069;
        double r24198092 = r24198089 - r24198091;
        double r24198093 = r24198074 ? r24198082 : r24198092;
        double r24198094 = r24198045 ? r24198072 : r24198093;
        return r24198094;
}

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

Bits error versus i

Bits error versus j

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if y < -1.7461996372679715e-56

    1. Initial program 9.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied *-un-lft-identity9.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot \color{blue}{\left(1 \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    4. Applied associate-*r*9.4

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot 1\right) \cdot t} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    5. Simplified6.2

      \[\leadsto \left(\left(\left(\color{blue}{\left(y \cdot \left(x \cdot \left(z \cdot 18.0\right)\right)\right)} \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    6. Using strategy rm

    7. Applied associate-*l*2.1

      \[\leadsto \left(\left(\left(\color{blue}{y \cdot \left(\left(x \cdot \left(z \cdot 18.0\right)\right) \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    8. Using strategy rm

    9. Applied associate-*l*1.8

      \[\leadsto \left(\left(\left(y \cdot \color{blue}{\left(x \cdot \left(\left(z \cdot 18.0\right) \cdot t\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    10. Using strategy rm

    11. Applied associate-*l*1.8

      \[\leadsto \left(\left(\left(y \cdot \left(x \cdot \color{blue}{\left(z \cdot \left(18.0 \cdot t\right)\right)}\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if -1.7461996372679715e-56 < y < 3.73999027982069e-174

    1. Initial program 0.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right), t, \left(c \cdot b - \mathsf{fma}\left(\left(4.0 \cdot i\right), x, \left(\left(27.0 \cdot j\right) \cdot k\right)\right)\right)\right)}\]

    if 3.73999027982069e-174 < y

    1. Initial program 6.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    2. Using strategy rm

    3. Applied *-un-lft-identity6.8

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot \color{blue}{\left(1 \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    4. Applied associate-*r*6.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot 1\right) \cdot t} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    5. Simplified5.9

      \[\leadsto \left(\left(\left(\color{blue}{\left(y \cdot \left(x \cdot \left(z \cdot 18.0\right)\right)\right)} \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    6. Using strategy rm

    7. Applied associate-*l*3.8

      \[\leadsto \left(\left(\left(\color{blue}{y \cdot \left(\left(x \cdot \left(z \cdot 18.0\right)\right) \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    8. Using strategy rm

    9. Applied associate-*l*3.7

      \[\leadsto \left(\left(\left(y \cdot \color{blue}{\left(x \cdot \left(\left(z \cdot 18.0\right) \cdot t\right)\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    10. Taylor expanded around -inf 3.5

      \[\leadsto \left(\left(\left(y \cdot \left(x \cdot \left(\left(z \cdot 18.0\right) \cdot t\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{27.0 \cdot \left(j \cdot k\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.7461996372679715 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(18.0 \cdot t\right) \cdot z\right) \cdot x\right) \cdot y - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;y \le 3.73999027982069 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right), t, \left(b \cdot c - \mathsf{fma}\left(\left(4.0 \cdot i\right), x, \left(k \cdot \left(j \cdot 27.0\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(x \cdot \left(t \cdot \left(z \cdot 18.0\right)\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))