Average Error: 11.4 → 8.7
Time: 23.6s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.5826701141979015 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(z \cdot y - t \cdot a\right)\right)\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\\ \mathbf{elif}\;b \le 6.450825945075627 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(z \cdot y - t \cdot a\right) \cdot x\right)\right) - \left(z \cdot \left(c \cdot b\right) - \left(i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(z \cdot y - t \cdot a\right) \cdot x\right)\right) - \left(\sqrt{b} \cdot \left(z \cdot c - i \cdot a\right)\right) \cdot \sqrt{b}\\ \end{array}\]
\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 -2.5826701141979015 \cdot 10^{+70}:\\
\;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(z \cdot y - t \cdot a\right)\right)\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\\

\mathbf{elif}\;b \le 6.450825945075627 \cdot 10^{-74}:\\
\;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(z \cdot y - t \cdot a\right) \cdot x\right)\right) - \left(z \cdot \left(c \cdot b\right) - \left(i \cdot b\right) \cdot a\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1775332 = x;
        double r1775333 = y;
        double r1775334 = z;
        double r1775335 = r1775333 * r1775334;
        double r1775336 = t;
        double r1775337 = a;
        double r1775338 = r1775336 * r1775337;
        double r1775339 = r1775335 - r1775338;
        double r1775340 = r1775332 * r1775339;
        double r1775341 = b;
        double r1775342 = c;
        double r1775343 = r1775342 * r1775334;
        double r1775344 = i;
        double r1775345 = r1775344 * r1775337;
        double r1775346 = r1775343 - r1775345;
        double r1775347 = r1775341 * r1775346;
        double r1775348 = r1775340 - r1775347;
        double r1775349 = j;
        double r1775350 = r1775342 * r1775336;
        double r1775351 = r1775344 * r1775333;
        double r1775352 = r1775350 - r1775351;
        double r1775353 = r1775349 * r1775352;
        double r1775354 = r1775348 + r1775353;
        return r1775354;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1775355 = b;
        double r1775356 = -2.5826701141979015e+70;
        bool r1775357 = r1775355 <= r1775356;
        double r1775358 = t;
        double r1775359 = c;
        double r1775360 = r1775358 * r1775359;
        double r1775361 = i;
        double r1775362 = y;
        double r1775363 = r1775361 * r1775362;
        double r1775364 = r1775360 - r1775363;
        double r1775365 = j;
        double r1775366 = x;
        double r1775367 = cbrt(r1775366);
        double r1775368 = r1775367 * r1775367;
        double r1775369 = z;
        double r1775370 = r1775369 * r1775362;
        double r1775371 = a;
        double r1775372 = r1775358 * r1775371;
        double r1775373 = r1775370 - r1775372;
        double r1775374 = r1775368 * r1775373;
        double r1775375 = r1775367 * r1775374;
        double r1775376 = fma(r1775364, r1775365, r1775375);
        double r1775377 = r1775369 * r1775359;
        double r1775378 = r1775361 * r1775371;
        double r1775379 = r1775377 - r1775378;
        double r1775380 = r1775355 * r1775379;
        double r1775381 = r1775376 - r1775380;
        double r1775382 = 6.450825945075627e-74;
        bool r1775383 = r1775355 <= r1775382;
        double r1775384 = r1775373 * r1775366;
        double r1775385 = fma(r1775364, r1775365, r1775384);
        double r1775386 = r1775359 * r1775355;
        double r1775387 = r1775369 * r1775386;
        double r1775388 = r1775361 * r1775355;
        double r1775389 = r1775388 * r1775371;
        double r1775390 = r1775387 - r1775389;
        double r1775391 = r1775385 - r1775390;
        double r1775392 = sqrt(r1775355);
        double r1775393 = r1775392 * r1775379;
        double r1775394 = r1775393 * r1775392;
        double r1775395 = r1775385 - r1775394;
        double r1775396 = r1775383 ? r1775391 : r1775395;
        double r1775397 = r1775357 ? r1775381 : r1775396;
        return r1775397;
}

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 b < -2.5826701141979015e+70

    1. Initial program 6.2

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

    2. Simplified6.2

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

    4. Applied add-cube-cbrt6.4

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

    5. Applied associate-*r*6.4

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

    if -2.5826701141979015e+70 < b < 6.450825945075627e-74

    1. Initial program 14.2

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

    2. Simplified14.2

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

    4. Applied add-cube-cbrt14.4

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

    5. Applied associate-*r*14.4

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

    6. Taylor expanded around -inf 9.5

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

    if 6.450825945075627e-74 < b

    1. Initial program 7.8

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

    2. Simplified7.8

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

    4. Applied add-sqr-sqrt7.9

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

    5. Applied associate-*r*7.9

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.5826701141979015 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(z \cdot y - t \cdot a\right)\right)\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\\ \mathbf{elif}\;b \le 6.450825945075627 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(z \cdot y - t \cdot a\right) \cdot x\right)\right) - \left(z \cdot \left(c \cdot b\right) - \left(i \cdot b\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(z \cdot y - t \cdot a\right) \cdot x\right)\right) - \left(\sqrt{b} \cdot \left(z \cdot c - i \cdot a\right)\right) \cdot \sqrt{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))))