Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.2 → 9.1

Time: 25.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.7988923798283882 \cdot 10^{+75}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \left(z \cdot c - a \cdot i\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\\ \mathbf{elif}\;b \le 1.205607812754414 \cdot 10^{+86}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - t \cdot a\right) - \left(\left(-i \cdot \left(b \cdot a\right)\right) + \left(c \cdot b\right) \cdot z\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \left(z \cdot c - a \cdot i\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r25430368 = x;
        double r25430369 = y;
        double r25430370 = z;
        double r25430371 = r25430369 * r25430370;
        double r25430372 = t;
        double r25430373 = a;
        double r25430374 = r25430372 * r25430373;
        double r25430375 = r25430371 - r25430374;
        double r25430376 = r25430368 * r25430375;
        double r25430377 = b;
        double r25430378 = c;
        double r25430379 = r25430378 * r25430370;
        double r25430380 = i;
        double r25430381 = r25430380 * r25430373;
        double r25430382 = r25430379 - r25430381;
        double r25430383 = r25430377 * r25430382;
        double r25430384 = r25430376 - r25430383;
        double r25430385 = j;
        double r25430386 = r25430378 * r25430372;
        double r25430387 = r25430380 * r25430369;
        double r25430388 = r25430386 - r25430387;
        double r25430389 = r25430385 * r25430388;
        double r25430390 = r25430384 + r25430389;
        return r25430390;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r25430391 = b;
        double r25430392 = -1.7988923798283882e+75;
        bool r25430393 = r25430391 <= r25430392;
        double r25430394 = j;
        double r25430395 = t;
        double r25430396 = c;
        double r25430397 = r25430395 * r25430396;
        double r25430398 = y;
        double r25430399 = i;
        double r25430400 = r25430398 * r25430399;
        double r25430401 = r25430397 - r25430400;
        double r25430402 = r25430394 * r25430401;
        double r25430403 = x;
        double r25430404 = z;
        double r25430405 = r25430404 * r25430398;
        double r25430406 = r25430403 * r25430405;
        double r25430407 = a;
        double r25430408 = r25430403 * r25430395;
        double r25430409 = r25430407 * r25430408;
        double r25430410 = r25430406 - r25430409;
        double r25430411 = cbrt(r25430391);
        double r25430412 = r25430404 * r25430396;
        double r25430413 = r25430407 * r25430399;
        double r25430414 = r25430412 - r25430413;
        double r25430415 = r25430411 * r25430414;
        double r25430416 = r25430411 * r25430411;
        double r25430417 = r25430415 * r25430416;
        double r25430418 = r25430410 - r25430417;
        double r25430419 = r25430402 + r25430418;
        double r25430420 = 1.205607812754414e+86;
        bool r25430421 = r25430391 <= r25430420;
        double r25430422 = r25430395 * r25430407;
        double r25430423 = r25430405 - r25430422;
        double r25430424 = r25430403 * r25430423;
        double r25430425 = r25430391 * r25430407;
        double r25430426 = r25430399 * r25430425;
        double r25430427 = -r25430426;
        double r25430428 = r25430396 * r25430391;
        double r25430429 = r25430428 * r25430404;
        double r25430430 = r25430427 + r25430429;
        double r25430431 = r25430424 - r25430430;
        double r25430432 = r25430431 + r25430402;
        double r25430433 = r25430421 ? r25430432 : r25430419;
        double r25430434 = r25430393 ? r25430419 : r25430433;
        return r25430434;
}

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

Target

Original	11.2
Target	15.1
Herbie	9.1

\[\begin{array}{l} \mathbf{if}\;t \lt -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if b < -1.7988923798283882e+75 or 1.205607812754414e+86 < b

   1. Initial program 6.4

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

   3. Applied add-cube-cbrt 6.9

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

   4. Applied associate-*l* 6.9

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

   5. Taylor expanded around inf 7.7

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

   if -1.7988923798283882e+75 < b < 1.205607812754414e+86

   1. Initial program 12.8

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

   3. Applied add-cube-cbrt 13.0

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

   4. Applied associate-*l* 13.0

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

   6. Applied sub-neg 13.0

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

   7. Applied distribute-lft-in 13.0

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

   8. Applied distribute-lft-in 13.0

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

   9. Simplified 11.3

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

   10. Simplified 9.5

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

   11. Taylor expanded around inf 9.5

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(c \cdot \left(z \cdot b\right) + \color{blue}{-1 \cdot \left(a \cdot \left(i \cdot b\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   12. Simplified 9.4

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

   13. Taylor expanded around inf 9.5

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{z \cdot \left(b \cdot c\right)} + \left(a \cdot b\right) \cdot \left(-i\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7988923798283882 \cdot 10^{+75}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \left(z \cdot c - a \cdot i\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\\ \mathbf{elif}\;b \le 1.205607812754414 \cdot 10^{+86}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - t \cdot a\right) - \left(\left(-i \cdot \left(b \cdot a\right)\right) + \left(c \cdot b\right) \cdot z\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \left(z \cdot c - a \cdot i\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))