Average Error: 12.7 → 12.5
Time: 9.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2063988814437769 \cdot 10^{-250}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;b \le 1.52878530329258907 \cdot 10^{-189}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\ \end{array}\]
\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}\;b \le -5.2063988814437769 \cdot 10^{-250}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\

\mathbf{elif}\;b \le 1.52878530329258907 \cdot 10^{-189}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - 0\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\

\end{array}
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y))));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double VAR;
	if ((b <= -5.206398881443777e-250)) {
		VAR = fma(((c * t) - (i * y)), j, (((cbrt((x * ((y * z) - (t * a)))) * (cbrt(x) * cbrt(((y * z) - (t * a))))) * cbrt((x * ((y * z) - (t * a))))) - (b * ((c * z) - (i * a)))));
	} else {
		double VAR_1;
		if ((b <= 1.528785303292589e-189)) {
			VAR_1 = fma(((c * t) - (i * y)), j, ((x * ((y * z) - (t * a))) - 0.0));
		} else {
			VAR_1 = fma(((c * t) - (i * y)), j, ((x * ((y * z) - (t * a))) - (sqrt(b) * (sqrt(b) * ((c * z) - (i * a))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -5.206398881443777e-250

    1. Initial program 11.8

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

    2. Simplified11.8

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

    4. Applied add-cube-cbrt12.1

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

    6. Applied cbrt-prod12.1

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

    if -5.206398881443777e-250 < b < 1.528785303292589e-189

    1. Initial program 19.6

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

    2. Simplified19.6

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

    3. Taylor expanded around 0 17.8

      \[\leadsto \mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{0}\right)\]

    if 1.528785303292589e-189 < b

    1. Initial program 10.4

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

    2. Simplified10.4

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

    4. Applied add-sqr-sqrt10.5

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

    5. Applied associate-*l*10.5

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2063988814437769 \cdot 10^{-250}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;b \le 1.52878530329258907 \cdot 10^{-189}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - 0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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)))))