Average Error: 5.4 → 4.8
Time: 17.3s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.8707220913250884 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(\left(\sqrt[3]{\left(x \cdot z\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot z\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot z\right) \cdot y}\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 4.82718688204051576 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]
\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 -3.8707220913250884 \cdot 10^{-78}:\\
\;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(\left(\sqrt[3]{\left(x \cdot z\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot z\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot z\right) \cdot y}\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 4.82718688204051576 \cdot 10^{-8}:\\
\;\;\;\;\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(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\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 r153020 = x;
        double r153021 = 18.0;
        double r153022 = r153020 * r153021;
        double r153023 = y;
        double r153024 = r153022 * r153023;
        double r153025 = z;
        double r153026 = r153024 * r153025;
        double r153027 = t;
        double r153028 = r153026 * r153027;
        double r153029 = a;
        double r153030 = 4.0;
        double r153031 = r153029 * r153030;
        double r153032 = r153031 * r153027;
        double r153033 = r153028 - r153032;
        double r153034 = b;
        double r153035 = c;
        double r153036 = r153034 * r153035;
        double r153037 = r153033 + r153036;
        double r153038 = r153020 * r153030;
        double r153039 = i;
        double r153040 = r153038 * r153039;
        double r153041 = r153037 - r153040;
        double r153042 = j;
        double r153043 = 27.0;
        double r153044 = r153042 * r153043;
        double r153045 = k;
        double r153046 = r153044 * r153045;
        double r153047 = r153041 - r153046;
        return r153047;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r153048 = t;
        double r153049 = -3.8707220913250884e-78;
        bool r153050 = r153048 <= r153049;
        double r153051 = 18.0;
        double r153052 = x;
        double r153053 = z;
        double r153054 = r153052 * r153053;
        double r153055 = y;
        double r153056 = r153054 * r153055;
        double r153057 = cbrt(r153056);
        double r153058 = r153057 * r153057;
        double r153059 = r153058 * r153057;
        double r153060 = r153051 * r153059;
        double r153061 = a;
        double r153062 = 4.0;
        double r153063 = r153061 * r153062;
        double r153064 = r153060 - r153063;
        double r153065 = b;
        double r153066 = c;
        double r153067 = r153065 * r153066;
        double r153068 = i;
        double r153069 = r153062 * r153068;
        double r153070 = j;
        double r153071 = 27.0;
        double r153072 = r153070 * r153071;
        double r153073 = k;
        double r153074 = r153072 * r153073;
        double r153075 = fma(r153052, r153069, r153074);
        double r153076 = r153067 - r153075;
        double r153077 = fma(r153048, r153064, r153076);
        double r153078 = 4.827186882040516e-08;
        bool r153079 = r153048 <= r153078;
        double r153080 = 0.0;
        double r153081 = r153080 - r153063;
        double r153082 = fma(r153048, r153081, r153076);
        double r153083 = r153052 * r153051;
        double r153084 = r153083 * r153055;
        double r153085 = r153084 * r153053;
        double r153086 = r153085 - r153063;
        double r153087 = r153071 * r153073;
        double r153088 = r153070 * r153087;
        double r153089 = fma(r153052, r153069, r153088);
        double r153090 = r153067 - r153089;
        double r153091 = fma(r153048, r153086, r153090);
        double r153092 = r153079 ? r153082 : r153091;
        double r153093 = r153050 ? r153077 : r153092;
        return r153093;
}

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 < -3.8707220913250884e-78

    1. Initial program 2.4

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

    2. Simplified2.5

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

    3. Taylor expanded around inf 3.2

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

    5. Applied associate-*r*2.3

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

    7. Applied add-cube-cbrt2.5

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

    if -3.8707220913250884e-78 < t < 4.827186882040516e-08

    1. Initial program 8.1

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

    2. Simplified8.2

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

    3. Taylor expanded around 0 6.9

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

    if 4.827186882040516e-08 < t

    1. Initial program 1.6

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

    2. Simplified1.8

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

    4. Applied associate-*l*1.8

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.8707220913250884 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(\left(\sqrt[3]{\left(x \cdot z\right) \cdot y} \cdot \sqrt[3]{\left(x \cdot z\right) \cdot y}\right) \cdot \sqrt[3]{\left(x \cdot z\right) \cdot y}\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;t \le 4.82718688204051576 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))