Average Error: 5.7 → 4.8
Time: 18.8s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.816104989340000263533513256286904752193 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 1.945396783892281696257471720471681450496 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.816104989340000263533513256286904752193 \cdot 10^{-100}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{elif}\;t \le 1.945396783892281696257471720471681450496 \cdot 10^{-140}:\\
\;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\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 r166069 = x;
        double r166070 = 18.0;
        double r166071 = r166069 * r166070;
        double r166072 = y;
        double r166073 = r166071 * r166072;
        double r166074 = z;
        double r166075 = r166073 * r166074;
        double r166076 = t;
        double r166077 = r166075 * r166076;
        double r166078 = a;
        double r166079 = 4.0;
        double r166080 = r166078 * r166079;
        double r166081 = r166080 * r166076;
        double r166082 = r166077 - r166081;
        double r166083 = b;
        double r166084 = c;
        double r166085 = r166083 * r166084;
        double r166086 = r166082 + r166085;
        double r166087 = r166069 * r166079;
        double r166088 = i;
        double r166089 = r166087 * r166088;
        double r166090 = r166086 - r166089;
        double r166091 = j;
        double r166092 = 27.0;
        double r166093 = r166091 * r166092;
        double r166094 = k;
        double r166095 = r166093 * r166094;
        double r166096 = r166090 - r166095;
        return r166096;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r166097 = t;
        double r166098 = -1.8161049893400003e-100;
        bool r166099 = r166097 <= r166098;
        double r166100 = x;
        double r166101 = 18.0;
        double r166102 = r166100 * r166101;
        double r166103 = y;
        double r166104 = z;
        double r166105 = r166103 * r166104;
        double r166106 = r166102 * r166105;
        double r166107 = a;
        double r166108 = 4.0;
        double r166109 = r166107 * r166108;
        double r166110 = r166106 - r166109;
        double r166111 = b;
        double r166112 = c;
        double r166113 = r166111 * r166112;
        double r166114 = i;
        double r166115 = r166108 * r166114;
        double r166116 = j;
        double r166117 = 27.0;
        double r166118 = r166116 * r166117;
        double r166119 = k;
        double r166120 = r166118 * r166119;
        double r166121 = fma(r166100, r166115, r166120);
        double r166122 = r166113 - r166121;
        double r166123 = fma(r166097, r166110, r166122);
        double r166124 = 1.9453967838922817e-140;
        bool r166125 = r166097 <= r166124;
        double r166126 = 0.0;
        double r166127 = r166126 - r166109;
        double r166128 = fma(r166097, r166127, r166122);
        double r166129 = r166101 * r166103;
        double r166130 = r166100 * r166129;
        double r166131 = r166130 * r166104;
        double r166132 = r166131 - r166109;
        double r166133 = fma(r166097, r166132, r166122);
        double r166134 = r166125 ? r166128 : r166133;
        double r166135 = r166099 ? r166123 : r166134;
        return r166135;
}

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 t < -1.8161049893400003e-100

    1. Initial program 3.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified3.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*3.7

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(x \cdot 18\right) \cdot \left(y \cdot z\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    if -1.8161049893400003e-100 < t < 1.9453967838922817e-140

    1. Initial program 8.9

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified8.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]

    3. Taylor expanded around 0 6.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    if 1.9453967838922817e-140 < t

    1. Initial program 3.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified3.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*3.9

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(x \cdot \left(18 \cdot y\right)\right)} \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.816104989340000263533513256286904752193 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 1.945396783892281696257471720471681450496 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))