Average Error: 11.5 → 11.8
Time: 50.6s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\mathsf{fma}\left(j, \left(\mathsf{fma}\left(t, c, \left(y \cdot \left(-i\right)\right)\right)\right), \left(x \cdot \left(z \cdot y - a \cdot t\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \sqrt[3]{b}\right)\right)\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)
↓
\mathsf{fma}\left(j, \left(\mathsf{fma}\left(t, c, \left(y \cdot \left(-i\right)\right)\right)\right), \left(x \cdot \left(z \cdot y - a \cdot t\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(c \cdot z - i \cdot a\right)\right) \cdot \sqrt[3]{b}\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r22037140 = x;
        double r22037141 = y;
        double r22037142 = z;
        double r22037143 = r22037141 * r22037142;
        double r22037144 = t;
        double r22037145 = a;
        double r22037146 = r22037144 * r22037145;
        double r22037147 = r22037143 - r22037146;
        double r22037148 = r22037140 * r22037147;
        double r22037149 = b;
        double r22037150 = c;
        double r22037151 = r22037150 * r22037142;
        double r22037152 = i;
        double r22037153 = r22037152 * r22037145;
        double r22037154 = r22037151 - r22037153;
        double r22037155 = r22037149 * r22037154;
        double r22037156 = r22037148 - r22037155;
        double r22037157 = j;
        double r22037158 = r22037150 * r22037144;
        double r22037159 = r22037152 * r22037141;
        double r22037160 = r22037158 - r22037159;
        double r22037161 = r22037157 * r22037160;
        double r22037162 = r22037156 + r22037161;
        return r22037162;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r22037163 = j;
        double r22037164 = t;
        double r22037165 = c;
        double r22037166 = y;
        double r22037167 = i;
        double r22037168 = -r22037167;
        double r22037169 = r22037166 * r22037168;
        double r22037170 = fma(r22037164, r22037165, r22037169);
        double r22037171 = x;
        double r22037172 = z;
        double r22037173 = r22037172 * r22037166;
        double r22037174 = a;
        double r22037175 = r22037174 * r22037164;
        double r22037176 = r22037173 - r22037175;
        double r22037177 = r22037171 * r22037176;
        double r22037178 = b;
        double r22037179 = cbrt(r22037178);
        double r22037180 = r22037179 * r22037179;
        double r22037181 = r22037165 * r22037172;
        double r22037182 = r22037167 * r22037174;
        double r22037183 = r22037181 - r22037182;
        double r22037184 = r22037180 * r22037183;
        double r22037185 = r22037184 * r22037179;
        double r22037186 = r22037177 - r22037185;
        double r22037187 = fma(r22037163, r22037170, r22037186);
        return r22037187;
}

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

Derivation

  1. Initial program 11.5

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

  2. Simplified11.5

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

  4. Applied fma-neg11.5

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

  6. Applied add-cube-cbrt11.8

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

  7. Applied associate-*r*11.8

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

  8. Final simplification11.8

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))