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

↓

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

\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

↓

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

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r736901 = x;
        double r736902 = 18.0;
        double r736903 = r736901 * r736902;
        double r736904 = y;
        double r736905 = r736903 * r736904;
        double r736906 = z;
        double r736907 = r736905 * r736906;
        double r736908 = t;
        double r736909 = r736907 * r736908;
        double r736910 = a;
        double r736911 = 4.0;
        double r736912 = r736910 * r736911;
        double r736913 = r736912 * r736908;
        double r736914 = r736909 - r736913;
        double r736915 = b;
        double r736916 = c;
        double r736917 = r736915 * r736916;
        double r736918 = r736914 + r736917;
        double r736919 = r736901 * r736911;
        double r736920 = i;
        double r736921 = r736919 * r736920;
        double r736922 = r736918 - r736921;
        double r736923 = j;
        double r736924 = 27.0;
        double r736925 = r736923 * r736924;
        double r736926 = k;
        double r736927 = r736925 * r736926;
        double r736928 = r736922 - r736927;
        return r736928;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r736929 = t;
        double r736930 = x;
        double r736931 = 18.0;
        double r736932 = r736930 * r736931;
        double r736933 = y;
        double r736934 = r736932 * r736933;
        double r736935 = z;
        double r736936 = cbrt(r736935);
        double r736937 = r736936 * r736936;
        double r736938 = r736934 * r736937;
        double r736939 = r736938 * r736936;
        double r736940 = a;
        double r736941 = 4.0;
        double r736942 = r736940 * r736941;
        double r736943 = r736939 - r736942;
        double r736944 = b;
        double r736945 = c;
        double r736946 = r736944 * r736945;
        double r736947 = i;
        double r736948 = r736941 * r736947;
        double r736949 = j;
        double r736950 = 27.0;
        double r736951 = k;
        double r736952 = r736950 * r736951;
        double r736953 = r736949 * r736952;
        double r736954 = fma(r736930, r736948, r736953);
        double r736955 = r736946 - r736954;
        double r736956 = fma(r736929, r736943, r736955);
        return r736956;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	5.6
Target	1.6
Herbie	5.7

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

Initial program 5.6
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
Simplified5.6
\[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
Using strategy rm
Applied associate-*l*5.6
\[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
Using strategy rm
Applied add-cube-cbrt5.7
\[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]
Applied associate-*r*5.7
\[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]
Final simplification5.7
\[\leadsto \mathsf{fma}\left(t, \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))