Average Error: 5.5 → 4.5
Time: 19.5s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.16781444686311266 \cdot 10^{-150} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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 -1.16781444686311266 \cdot 10^{-150} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\
\;\;\;\;\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(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\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 r161023 = x;
        double r161024 = 18.0;
        double r161025 = r161023 * r161024;
        double r161026 = y;
        double r161027 = r161025 * r161026;
        double r161028 = z;
        double r161029 = r161027 * r161028;
        double r161030 = t;
        double r161031 = r161029 * r161030;
        double r161032 = a;
        double r161033 = 4.0;
        double r161034 = r161032 * r161033;
        double r161035 = r161034 * r161030;
        double r161036 = r161031 - r161035;
        double r161037 = b;
        double r161038 = c;
        double r161039 = r161037 * r161038;
        double r161040 = r161036 + r161039;
        double r161041 = r161023 * r161033;
        double r161042 = i;
        double r161043 = r161041 * r161042;
        double r161044 = r161040 - r161043;
        double r161045 = j;
        double r161046 = 27.0;
        double r161047 = r161045 * r161046;
        double r161048 = k;
        double r161049 = r161047 * r161048;
        double r161050 = r161044 - r161049;
        return r161050;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r161051 = t;
        double r161052 = -1.1678144468631127e-150;
        bool r161053 = r161051 <= r161052;
        double r161054 = 4.1450304107497156e-137;
        bool r161055 = r161051 <= r161054;
        double r161056 = !r161055;
        bool r161057 = r161053 || r161056;
        double r161058 = x;
        double r161059 = 18.0;
        double r161060 = r161058 * r161059;
        double r161061 = y;
        double r161062 = r161060 * r161061;
        double r161063 = z;
        double r161064 = r161062 * r161063;
        double r161065 = a;
        double r161066 = 4.0;
        double r161067 = r161065 * r161066;
        double r161068 = r161064 - r161067;
        double r161069 = b;
        double r161070 = c;
        double r161071 = r161069 * r161070;
        double r161072 = i;
        double r161073 = r161066 * r161072;
        double r161074 = j;
        double r161075 = 27.0;
        double r161076 = r161074 * r161075;
        double r161077 = k;
        double r161078 = cbrt(r161077);
        double r161079 = r161078 * r161078;
        double r161080 = r161076 * r161079;
        double r161081 = r161080 * r161078;
        double r161082 = fma(r161058, r161073, r161081);
        double r161083 = r161071 - r161082;
        double r161084 = fma(r161051, r161068, r161083);
        double r161085 = 0.0;
        double r161086 = r161085 - r161067;
        double r161087 = r161076 * r161077;
        double r161088 = fma(r161058, r161073, r161087);
        double r161089 = r161071 - r161088;
        double r161090 = fma(r161051, r161086, r161089);
        double r161091 = r161057 ? r161084 : r161090;
        return r161091;
}

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 2 regimes

  2. if t < -1.1678144468631127e-150 or 4.1450304107497156e-137 < t

    1. Initial program 3.4

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

    2. Simplified3.4

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

    4. Applied add-cube-cbrt3.6

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

    5. Applied associate-*r*3.6

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

    if -1.1678144468631127e-150 < t < 4.1450304107497156e-137

    1. Initial program 9.6

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

    2. Simplified9.6

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

    3. Taylor expanded around 0 6.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.16781444686311266 \cdot 10^{-150} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))