Average Error: 12.1 → 12.1
Time: 8.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.88745775492739174 \cdot 10^{-157}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{elif}\;x \le 3.1985989822224201 \cdot 10^{-251}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]
\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 -2.88745775492739174 \cdot 10^{-157}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\

\mathbf{elif}\;x \le 3.1985989822224201 \cdot 10^{-251}:\\
\;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r139203 = x;
        double r139204 = y;
        double r139205 = z;
        double r139206 = r139204 * r139205;
        double r139207 = t;
        double r139208 = a;
        double r139209 = r139207 * r139208;
        double r139210 = r139206 - r139209;
        double r139211 = r139203 * r139210;
        double r139212 = b;
        double r139213 = c;
        double r139214 = r139213 * r139205;
        double r139215 = i;
        double r139216 = r139215 * r139208;
        double r139217 = r139214 - r139216;
        double r139218 = r139212 * r139217;
        double r139219 = r139211 - r139218;
        double r139220 = j;
        double r139221 = r139213 * r139207;
        double r139222 = r139215 * r139204;
        double r139223 = r139221 - r139222;
        double r139224 = r139220 * r139223;
        double r139225 = r139219 + r139224;
        return r139225;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r139226 = x;
        double r139227 = -2.8874577549273917e-157;
        bool r139228 = r139226 <= r139227;
        double r139229 = y;
        double r139230 = z;
        double r139231 = r139229 * r139230;
        double r139232 = t;
        double r139233 = a;
        double r139234 = r139232 * r139233;
        double r139235 = r139231 - r139234;
        double r139236 = r139226 * r139235;
        double r139237 = b;
        double r139238 = c;
        double r139239 = r139238 * r139230;
        double r139240 = i;
        double r139241 = r139240 * r139233;
        double r139242 = r139239 - r139241;
        double r139243 = r139237 * r139242;
        double r139244 = -r139233;
        double r139245 = r139233 * r139240;
        double r139246 = fma(r139244, r139240, r139245);
        double r139247 = r139237 * r139246;
        double r139248 = r139243 + r139247;
        double r139249 = r139236 - r139248;
        double r139250 = j;
        double r139251 = r139238 * r139232;
        double r139252 = r139240 * r139229;
        double r139253 = r139251 - r139252;
        double r139254 = cbrt(r139253);
        double r139255 = r139254 * r139254;
        double r139256 = r139250 * r139255;
        double r139257 = r139256 * r139254;
        double r139258 = r139249 + r139257;
        double r139259 = 3.19859898222242e-251;
        bool r139260 = r139226 <= r139259;
        double r139261 = 0.0;
        double r139262 = r139261 - r139243;
        double r139263 = r139250 * r139253;
        double r139264 = r139262 + r139263;
        double r139265 = sqrt(r139226);
        double r139266 = r139265 * r139235;
        double r139267 = r139265 * r139266;
        double r139268 = r139267 - r139248;
        double r139269 = r139268 + r139263;
        double r139270 = r139260 ? r139264 : r139269;
        double r139271 = r139228 ? r139258 : r139270;
        return r139271;
}

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 < -2.8874577549273917e-157

    1. Initial program 9.3

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

    3. Applied prod-diff9.3

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

    4. Applied distribute-lft-in9.3

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

    5. Simplified9.3

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

    7. Applied add-cube-cbrt9.6

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

    8. Applied associate-*r*9.6

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

    if -2.8874577549273917e-157 < x < 3.19859898222242e-251

    1. Initial program 17.7

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

    2. Taylor expanded around 0 17.2

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

    if 3.19859898222242e-251 < x

    1. Initial program 11.4

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

    3. Applied prod-diff11.4

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

    4. Applied distribute-lft-in11.4

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

    5. Simplified11.4

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

    7. Applied add-sqr-sqrt11.5

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

    8. Applied associate-*l*11.5

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.88745775492739174 \cdot 10^{-157}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{elif}\;x \le 3.1985989822224201 \cdot 10^{-251}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))))