Average Error: 12.4 → 11.8
Time: 9.2s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 5.1580528935561765 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\right)\\ \end{array}\]
\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)
↓
\begin{array}{l}
\mathbf{if}\;a \le 5.1580528935561765 \cdot 10^{-13}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\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 r143056 = x;
        double r143057 = y;
        double r143058 = z;
        double r143059 = r143057 * r143058;
        double r143060 = t;
        double r143061 = a;
        double r143062 = r143060 * r143061;
        double r143063 = r143059 - r143062;
        double r143064 = r143056 * r143063;
        double r143065 = b;
        double r143066 = c;
        double r143067 = r143066 * r143058;
        double r143068 = i;
        double r143069 = r143068 * r143061;
        double r143070 = r143067 - r143069;
        double r143071 = r143065 * r143070;
        double r143072 = r143064 - r143071;
        double r143073 = j;
        double r143074 = r143066 * r143060;
        double r143075 = r143068 * r143057;
        double r143076 = r143074 - r143075;
        double r143077 = r143073 * r143076;
        double r143078 = r143072 + r143077;
        return r143078;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r143079 = a;
        double r143080 = 5.158052893556176e-13;
        bool r143081 = r143079 <= r143080;
        double r143082 = c;
        double r143083 = t;
        double r143084 = r143082 * r143083;
        double r143085 = i;
        double r143086 = y;
        double r143087 = r143085 * r143086;
        double r143088 = r143084 - r143087;
        double r143089 = j;
        double r143090 = x;
        double r143091 = z;
        double r143092 = r143086 * r143091;
        double r143093 = r143083 * r143079;
        double r143094 = r143092 - r143093;
        double r143095 = r143090 * r143094;
        double r143096 = b;
        double r143097 = r143082 * r143091;
        double r143098 = r143085 * r143079;
        double r143099 = r143097 - r143098;
        double r143100 = cbrt(r143099);
        double r143101 = r143100 * r143100;
        double r143102 = r143096 * r143101;
        double r143103 = r143102 * r143100;
        double r143104 = r143095 - r143103;
        double r143105 = fma(r143088, r143089, r143104);
        double r143106 = r143085 * r143096;
        double r143107 = r143096 * r143082;
        double r143108 = r143090 * r143083;
        double r143109 = r143079 * r143108;
        double r143110 = fma(r143091, r143107, r143109);
        double r143111 = -r143110;
        double r143112 = fma(r143079, r143106, r143111);
        double r143113 = fma(r143088, r143089, r143112);
        double r143114 = r143081 ? r143105 : r143113;
        return r143114;
}

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. Split input into 2 regimes

  2. if a < 5.158052893556176e-13

    1. Initial program 11.3

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

    2. Simplified11.3

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

    4. Applied add-cube-cbrt11.5

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

    5. Applied associate-*r*11.5

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

    if 5.158052893556176e-13 < a

    1. Initial program 16.8

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

    2. Simplified16.8

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

    3. Taylor expanded around inf 12.6

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

    4. Simplified12.6

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 5.1580528935561765 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))