Average Error: 3.8 → 0.4
Time: 20.3s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le -1.108112517456829553776834541471009972451 \cdot 10^{301}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(t \cdot \left(z \cdot 9\right)\right)\right)\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 3.292770907690504531217751539307706061638 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27 - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(t \cdot \left(z \cdot 9\right)\right)\right)\\ \end{array}\]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le -1.108112517456829553776834541471009972451 \cdot 10^{301}:\\
\;\;\;\;\mathsf{fma}\left(2, x, \sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(t \cdot \left(z \cdot 9\right)\right)\right)\\

\mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 3.292770907690504531217751539307706061638 \cdot 10^{305}:\\
\;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27 - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(2, x, \sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(t \cdot \left(z \cdot 9\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r32295048 = x;
        double r32295049 = 2.0;
        double r32295050 = r32295048 * r32295049;
        double r32295051 = y;
        double r32295052 = 9.0;
        double r32295053 = r32295051 * r32295052;
        double r32295054 = z;
        double r32295055 = r32295053 * r32295054;
        double r32295056 = t;
        double r32295057 = r32295055 * r32295056;
        double r32295058 = r32295050 - r32295057;
        double r32295059 = a;
        double r32295060 = 27.0;
        double r32295061 = r32295059 * r32295060;
        double r32295062 = b;
        double r32295063 = r32295061 * r32295062;
        double r32295064 = r32295058 + r32295063;
        return r32295064;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r32295065 = y;
        double r32295066 = 9.0;
        double r32295067 = r32295065 * r32295066;
        double r32295068 = z;
        double r32295069 = r32295067 * r32295068;
        double r32295070 = t;
        double r32295071 = r32295069 * r32295070;
        double r32295072 = -1.1081125174568296e+301;
        bool r32295073 = r32295071 <= r32295072;
        double r32295074 = 2.0;
        double r32295075 = x;
        double r32295076 = a;
        double r32295077 = 27.0;
        double r32295078 = b;
        double r32295079 = r32295077 * r32295078;
        double r32295080 = r32295076 * r32295079;
        double r32295081 = cbrt(r32295080);
        double r32295082 = r32295081 * r32295081;
        double r32295083 = r32295081 * r32295082;
        double r32295084 = r32295068 * r32295066;
        double r32295085 = r32295070 * r32295084;
        double r32295086 = r32295065 * r32295085;
        double r32295087 = r32295083 - r32295086;
        double r32295088 = fma(r32295074, r32295075, r32295087);
        double r32295089 = 3.2927709076905045e+305;
        bool r32295090 = r32295071 <= r32295089;
        double r32295091 = r32295078 * r32295076;
        double r32295092 = r32295091 * r32295077;
        double r32295093 = r32295068 * r32295065;
        double r32295094 = r32295070 * r32295093;
        double r32295095 = r32295066 * r32295094;
        double r32295096 = r32295092 - r32295095;
        double r32295097 = fma(r32295074, r32295075, r32295096);
        double r32295098 = r32295090 ? r32295097 : r32295088;
        double r32295099 = r32295073 ? r32295088 : r32295098;
        return r32295099;
}

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

Target

Original3.8
Target2.8
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* y 9.0) z) t) < -1.1081125174568296e+301 or 3.2927709076905045e+305 < (* (* (* y 9.0) z) t)

    1. Initial program 60.2

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified60.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*2.5

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-*l*1.5

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \color{blue}{y \cdot \left(9 \cdot \left(z \cdot t\right)\right)}\right)\]
    7. Using strategy rm

    8. Applied associate-*r*2.1

      \[\leadsto \mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - y \cdot \color{blue}{\left(\left(9 \cdot z\right) \cdot t\right)}\right)\]
    9. Using strategy rm

    10. Applied add-cube-cbrt2.2

      \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{\left(\sqrt[3]{\left(27 \cdot b\right) \cdot a} \cdot \sqrt[3]{\left(27 \cdot b\right) \cdot a}\right) \cdot \sqrt[3]{\left(27 \cdot b\right) \cdot a}} - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\]

    if -1.1081125174568296e+301 < (* (* (* y 9.0) z) t) < 3.2927709076905045e+305

    1. Initial program 0.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \left(27 \cdot b\right) \cdot a - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]

    3. Taylor expanded around inf 0.3

      \[\leadsto \mathsf{fma}\left(2, x, \color{blue}{27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le -1.108112517456829553776834541471009972451 \cdot 10^{301}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(t \cdot \left(z \cdot 9\right)\right)\right)\\ \mathbf{elif}\;\left(\left(y \cdot 9\right) \cdot z\right) \cdot t \le 3.292770907690504531217751539307706061638 \cdot 10^{305}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \left(b \cdot a\right) \cdot 27 - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \left(\sqrt[3]{a \cdot \left(27 \cdot b\right)} \cdot \sqrt[3]{a \cdot \left(27 \cdot b\right)}\right) - y \cdot \left(t \cdot \left(z \cdot 9\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))