Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 12.0

Time: 25.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.596778456688688 \cdot 10^{+165}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;z \le 1.0422704831254364 \cdot 10^{-259}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\ \mathbf{elif}\;z \le 2.2139058582803834 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;z \le 3.900370725857475 \cdot 10^{+273}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - \left(t \cdot x\right) \cdot a\right) - \left(i \cdot \left(-b \cdot a\right) + z \cdot \left(c \cdot b\right)\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r29862662 = x;
        double r29862663 = y;
        double r29862664 = z;
        double r29862665 = r29862663 * r29862664;
        double r29862666 = t;
        double r29862667 = a;
        double r29862668 = r29862666 * r29862667;
        double r29862669 = r29862665 - r29862668;
        double r29862670 = r29862662 * r29862669;
        double r29862671 = b;
        double r29862672 = c;
        double r29862673 = r29862672 * r29862664;
        double r29862674 = i;
        double r29862675 = r29862674 * r29862667;
        double r29862676 = r29862673 - r29862675;
        double r29862677 = r29862671 * r29862676;
        double r29862678 = r29862670 - r29862677;
        double r29862679 = j;
        double r29862680 = r29862672 * r29862666;
        double r29862681 = r29862674 * r29862663;
        double r29862682 = r29862680 - r29862681;
        double r29862683 = r29862679 * r29862682;
        double r29862684 = r29862678 + r29862683;
        return r29862684;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r29862685 = z;
        double r29862686 = -3.596778456688688e+165;
        bool r29862687 = r29862685 <= r29862686;
        double r29862688 = x;
        double r29862689 = r29862688 * r29862685;
        double r29862690 = y;
        double r29862691 = r29862689 * r29862690;
        double r29862692 = t;
        double r29862693 = r29862692 * r29862688;
        double r29862694 = a;
        double r29862695 = r29862693 * r29862694;
        double r29862696 = r29862691 - r29862695;
        double r29862697 = b;
        double r29862698 = c;
        double r29862699 = r29862698 * r29862685;
        double r29862700 = i;
        double r29862701 = r29862700 * r29862694;
        double r29862702 = r29862699 - r29862701;
        double r29862703 = r29862697 * r29862702;
        double r29862704 = r29862696 - r29862703;
        double r29862705 = r29862698 * r29862692;
        double r29862706 = r29862700 * r29862690;
        double r29862707 = r29862705 - r29862706;
        double r29862708 = j;
        double r29862709 = r29862707 * r29862708;
        double r29862710 = r29862704 + r29862709;
        double r29862711 = 1.0422704831254364e-259;
        bool r29862712 = r29862685 <= r29862711;
        double r29862713 = r29862690 * r29862685;
        double r29862714 = r29862694 * r29862692;
        double r29862715 = r29862713 - r29862714;
        double r29862716 = r29862715 * r29862688;
        double r29862717 = cbrt(r29862702);
        double r29862718 = r29862717 * r29862717;
        double r29862719 = r29862697 * r29862718;
        double r29862720 = r29862719 * r29862717;
        double r29862721 = r29862716 - r29862720;
        double r29862722 = r29862709 + r29862721;
        double r29862723 = 2.2139058582803834e+221;
        bool r29862724 = r29862685 <= r29862723;
        double r29862725 = 3.900370725857475e+273;
        bool r29862726 = r29862685 <= r29862725;
        double r29862727 = r29862713 * r29862688;
        double r29862728 = r29862727 - r29862695;
        double r29862729 = r29862697 * r29862694;
        double r29862730 = -r29862729;
        double r29862731 = r29862700 * r29862730;
        double r29862732 = r29862698 * r29862697;
        double r29862733 = r29862685 * r29862732;
        double r29862734 = r29862731 + r29862733;
        double r29862735 = r29862728 - r29862734;
        double r29862736 = r29862735 + r29862709;
        double r29862737 = r29862726 ? r29862736 : r29862710;
        double r29862738 = r29862724 ? r29862710 : r29862737;
        double r29862739 = r29862712 ? r29862722 : r29862738;
        double r29862740 = r29862687 ? r29862710 : r29862739;
        return r29862740;
}

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	12.1
Target	16.2
Herbie	12.0

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

Derivation

1. Split input into 3 regimes

2. if z < -3.596778456688688e+165 or 1.0422704831254364e-259 < z < 2.2139058582803834e+221 or 3.900370725857475e+273 < z

   1. Initial program 13.4

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

   3. Applied add-cube-cbrt 13.7

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

   4. Applied associate-*l* 13.7

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

   5. Taylor expanded around inf 13.7

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

   7. Applied associate-*r* 13.7

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

   9. Applied associate-*r* 13.7

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

   10. Simplified 13.4

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

   if -3.596778456688688e+165 < z < 1.0422704831254364e-259

   1. Initial program 10.3

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

   3. Applied add-cube-cbrt 10.6

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

   4. Applied associate-*r* 10.6

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

   if 2.2139058582803834e+221 < z < 3.900370725857475e+273

   1. Initial program 19.5

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

   3. Applied add-cube-cbrt 19.8

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

   4. Applied associate-*l* 19.8

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

   5. Taylor expanded around inf 19.8

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

   7. Applied sub-neg 19.8

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

   8. Applied distribute-lft-in 19.8

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

   9. Applied distribute-lft-in 19.8

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

   10. Simplified 14.4

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

   11. Simplified 12.4

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.596778456688688 \cdot 10^{+165}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;z \le 1.0422704831254364 \cdot 10^{-259}:\\ \;\;\;\;\left(c \cdot t - i \cdot y\right) \cdot j + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\ \mathbf{elif}\;z \le 2.2139058582803834 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{elif}\;z \le 3.900370725857475 \cdot 10^{+273}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - \left(t \cdot x\right) \cdot a\right) - \left(i \cdot \left(-b \cdot a\right) + z \cdot \left(c \cdot b\right)\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(c \cdot t - i \cdot y\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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)))))