Average Error: 12.2 → 12.2
Time: 29.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.732618327639122542554636862127635323692 \cdot 10^{-237}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, t, a \cdot t\right) \cdot x + \mathsf{fma}\left(y, z, \left(-a\right) \cdot t\right) \cdot x\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(t \cdot c - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;x \le 2.714727205484543614860096252924453206473 \cdot 10^{-192}:\\ \;\;\;\;\left(c \cdot z - a \cdot i\right) \cdot \left(-b\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \end{array}\]
\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}\;x \le -4.732618327639122542554636862127635323692 \cdot 10^{-237}:\\
\;\;\;\;\left(\left(\mathsf{fma}\left(-a, t, a \cdot t\right) \cdot x + \mathsf{fma}\left(y, z, \left(-a\right) \cdot t\right) \cdot x\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(t \cdot c - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\

\mathbf{elif}\;x \le 2.714727205484543614860096252924453206473 \cdot 10^{-192}:\\
\;\;\;\;\left(c \cdot z - a \cdot i\right) \cdot \left(-b\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\

\mathbf{else}:\\
\;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(c \cdot z - a \cdot i\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 r4725181 = x;
        double r4725182 = y;
        double r4725183 = z;
        double r4725184 = r4725182 * r4725183;
        double r4725185 = t;
        double r4725186 = a;
        double r4725187 = r4725185 * r4725186;
        double r4725188 = r4725184 - r4725187;
        double r4725189 = r4725181 * r4725188;
        double r4725190 = b;
        double r4725191 = c;
        double r4725192 = r4725191 * r4725183;
        double r4725193 = i;
        double r4725194 = r4725193 * r4725186;
        double r4725195 = r4725192 - r4725194;
        double r4725196 = r4725190 * r4725195;
        double r4725197 = r4725189 - r4725196;
        double r4725198 = j;
        double r4725199 = r4725191 * r4725185;
        double r4725200 = r4725193 * r4725182;
        double r4725201 = r4725199 - r4725200;
        double r4725202 = r4725198 * r4725201;
        double r4725203 = r4725197 + r4725202;
        return r4725203;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r4725204 = x;
        double r4725205 = -4.7326183276391225e-237;
        bool r4725206 = r4725204 <= r4725205;
        double r4725207 = a;
        double r4725208 = -r4725207;
        double r4725209 = t;
        double r4725210 = r4725207 * r4725209;
        double r4725211 = fma(r4725208, r4725209, r4725210);
        double r4725212 = r4725211 * r4725204;
        double r4725213 = y;
        double r4725214 = z;
        double r4725215 = r4725208 * r4725209;
        double r4725216 = fma(r4725213, r4725214, r4725215);
        double r4725217 = r4725216 * r4725204;
        double r4725218 = r4725212 + r4725217;
        double r4725219 = b;
        double r4725220 = c;
        double r4725221 = r4725220 * r4725214;
        double r4725222 = i;
        double r4725223 = r4725207 * r4725222;
        double r4725224 = r4725221 - r4725223;
        double r4725225 = r4725219 * r4725224;
        double r4725226 = r4725218 - r4725225;
        double r4725227 = j;
        double r4725228 = cbrt(r4725227);
        double r4725229 = r4725228 * r4725228;
        double r4725230 = r4725209 * r4725220;
        double r4725231 = r4725213 * r4725222;
        double r4725232 = r4725230 - r4725231;
        double r4725233 = r4725232 * r4725228;
        double r4725234 = r4725229 * r4725233;
        double r4725235 = r4725226 + r4725234;
        double r4725236 = 2.7147272054845436e-192;
        bool r4725237 = r4725204 <= r4725236;
        double r4725238 = -r4725219;
        double r4725239 = r4725224 * r4725238;
        double r4725240 = r4725227 * r4725232;
        double r4725241 = r4725239 + r4725240;
        double r4725242 = sqrt(r4725204);
        double r4725243 = r4725213 * r4725214;
        double r4725244 = r4725243 - r4725210;
        double r4725245 = r4725244 * r4725242;
        double r4725246 = r4725242 * r4725245;
        double r4725247 = r4725246 - r4725225;
        double r4725248 = r4725240 + r4725247;
        double r4725249 = r4725237 ? r4725241 : r4725248;
        double r4725250 = r4725206 ? r4725235 : r4725249;
        return r4725250;
}

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 3 regimes

  2. if x < -4.7326183276391225e-237

    1. Initial program 11.1

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    2. Using strategy rm

    3. Applied prod-diff11.1

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

    4. Applied distribute-lft-in11.1

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

    6. Applied add-cube-cbrt11.4

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

    7. Applied associate-*l*11.4

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

    if -4.7326183276391225e-237 < x < 2.7147272054845436e-192

    1. Initial program 17.4

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

    2. Taylor expanded around 0 16.8

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

    if 2.7147272054845436e-192 < x

    1. Initial program 10.6

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt10.7

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

    4. Applied associate-*l*10.7

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

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.732618327639122542554636862127635323692 \cdot 10^{-237}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-a, t, a \cdot t\right) \cdot x + \mathsf{fma}\left(y, z, \left(-a\right) \cdot t\right) \cdot x\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(t \cdot c - y \cdot i\right) \cdot \sqrt[3]{j}\right)\\ \mathbf{elif}\;x \le 2.714727205484543614860096252924453206473 \cdot 10^{-192}:\\ \;\;\;\;\left(c \cdot z - a \cdot i\right) \cdot \left(-b\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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)))))