Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 12.0

Time: 19.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -7.276076652476422723184151660607198493295 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;y \le -2.765207802801297782227349629851777181967 \cdot 10^{-96}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \le 7.448778321262618814210345647292621144017 \cdot 10^{-301}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;y \le 1.714973672352425960028893737750560179991 \cdot 10^{93}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r95450 = x;
        double r95451 = y;
        double r95452 = z;
        double r95453 = r95451 * r95452;
        double r95454 = t;
        double r95455 = a;
        double r95456 = r95454 * r95455;
        double r95457 = r95453 - r95456;
        double r95458 = r95450 * r95457;
        double r95459 = b;
        double r95460 = c;
        double r95461 = r95460 * r95452;
        double r95462 = i;
        double r95463 = r95462 * r95455;
        double r95464 = r95461 - r95463;
        double r95465 = r95459 * r95464;
        double r95466 = r95458 - r95465;
        double r95467 = j;
        double r95468 = r95460 * r95454;
        double r95469 = r95462 * r95451;
        double r95470 = r95468 - r95469;
        double r95471 = r95467 * r95470;
        double r95472 = r95466 + r95471;
        return r95472;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r95473 = y;
        double r95474 = -7.276076652476423e-42;
        bool r95475 = r95473 <= r95474;
        double r95476 = x;
        double r95477 = z;
        double r95478 = r95473 * r95477;
        double r95479 = r95476 * r95478;
        double r95480 = a;
        double r95481 = t;
        double r95482 = r95476 * r95481;
        double r95483 = r95480 * r95482;
        double r95484 = -r95483;
        double r95485 = r95479 + r95484;
        double r95486 = b;
        double r95487 = c;
        double r95488 = r95487 * r95477;
        double r95489 = i;
        double r95490 = r95489 * r95480;
        double r95491 = r95488 - r95490;
        double r95492 = cbrt(r95491);
        double r95493 = r95492 * r95492;
        double r95494 = r95486 * r95493;
        double r95495 = r95494 * r95492;
        double r95496 = r95485 - r95495;
        double r95497 = j;
        double r95498 = r95487 * r95481;
        double r95499 = r95489 * r95473;
        double r95500 = r95498 - r95499;
        double r95501 = r95497 * r95500;
        double r95502 = r95496 + r95501;
        double r95503 = -2.7652078028012978e-96;
        bool r95504 = r95473 <= r95503;
        double r95505 = r95481 * r95480;
        double r95506 = r95478 - r95505;
        double r95507 = r95476 * r95506;
        double r95508 = r95486 * r95491;
        double r95509 = r95507 - r95508;
        double r95510 = r95497 * r95487;
        double r95511 = r95481 * r95510;
        double r95512 = r95497 * r95473;
        double r95513 = r95489 * r95512;
        double r95514 = -r95513;
        double r95515 = r95511 + r95514;
        double r95516 = r95509 + r95515;
        double r95517 = 7.448778321262619e-301;
        bool r95518 = r95473 <= r95517;
        double r95519 = cbrt(r95476);
        double r95520 = r95519 * r95519;
        double r95521 = r95519 * r95478;
        double r95522 = r95520 * r95521;
        double r95523 = r95522 + r95484;
        double r95524 = r95486 * r95487;
        double r95525 = r95477 * r95524;
        double r95526 = -r95490;
        double r95527 = r95526 * r95486;
        double r95528 = r95525 + r95527;
        double r95529 = r95523 - r95528;
        double r95530 = r95529 + r95501;
        double r95531 = 1.714973672352426e+93;
        bool r95532 = r95473 <= r95531;
        double r95533 = r95476 * r95480;
        double r95534 = r95481 * r95533;
        double r95535 = -r95534;
        double r95536 = r95479 + r95535;
        double r95537 = r95536 - r95508;
        double r95538 = r95537 + r95501;
        double r95539 = r95532 ? r95516 : r95538;
        double r95540 = r95518 ? r95530 : r95539;
        double r95541 = r95504 ? r95516 : r95540;
        double r95542 = r95475 ? r95502 : r95541;
        return r95542;
}

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 4 regimes

2. if y < -7.276076652476423e-42

   1. Initial program 15.1

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

   3. Applied sub-neg 15.1

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

   4. Applied distribute-lft-in 15.1

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

   5. Simplified 14.7

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

   7. Applied add-cube-cbrt 14.9

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

   8. Applied associate-*r* 14.9

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

   if -7.276076652476423e-42 < y < -2.7652078028012978e-96 or 7.448778321262619e-301 < y < 1.714973672352426e+93

   1. Initial program 9.0

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

   3. Applied sub-neg 9.0

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

   4. Applied distribute-lft-in 9.0

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

   5. Simplified 9.6

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

   6. Simplified 9.5

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

   if -2.7652078028012978e-96 < y < 7.448778321262619e-301

   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. Using strategy rm

   3. Applied sub-neg 10.4

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

   4. Applied distribute-lft-in 10.4

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

   5. Simplified 10.4

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

   7. Applied add-cube-cbrt 10.5

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

   8. Applied associate-*l* 10.5

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

   10. Applied sub-neg 10.5

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

   11. Applied distribute-lft-in 10.5

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

   12. Simplified 9.8

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

   13. Simplified 9.8

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

   if 1.714973672352426e+93 < y

   1. Initial program 20.8

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

   3. Applied sub-neg 20.8

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

   4. Applied distribute-lft-in 20.8

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

   5. Simplified 20.4

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

   6. Taylor expanded around inf 19.9

      \[\leadsto \left(\left(x \cdot \left(y \cdot z\right) + \left(-\color{blue}{t \cdot \left(x \cdot a\right)}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.276076652476422723184151660607198493295 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;y \le -2.765207802801297782227349629851777181967 \cdot 10^{-96}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;y \le 7.448778321262618814210345647292621144017 \cdot 10^{-301}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;y \le 1.714973672352425960028893737750560179991 \cdot 10^{93}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))))