Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 12.7

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.0773273627394588 \cdot 10^{-176}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 3.74383731163816427 \cdot 10^{-169}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r2044580 = x;
        double r2044581 = y;
        double r2044582 = z;
        double r2044583 = r2044581 * r2044582;
        double r2044584 = t;
        double r2044585 = a;
        double r2044586 = r2044584 * r2044585;
        double r2044587 = r2044583 - r2044586;
        double r2044588 = r2044580 * r2044587;
        double r2044589 = b;
        double r2044590 = c;
        double r2044591 = r2044590 * r2044582;
        double r2044592 = i;
        double r2044593 = r2044592 * r2044585;
        double r2044594 = r2044591 - r2044593;
        double r2044595 = r2044589 * r2044594;
        double r2044596 = r2044588 - r2044595;
        double r2044597 = j;
        double r2044598 = r2044590 * r2044584;
        double r2044599 = r2044592 * r2044581;
        double r2044600 = r2044598 - r2044599;
        double r2044601 = r2044597 * r2044600;
        double r2044602 = r2044596 + r2044601;
        return r2044602;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r2044603 = x;
        double r2044604 = -3.0773273627394588e-176;
        bool r2044605 = r2044603 <= r2044604;
        double r2044606 = y;
        double r2044607 = z;
        double r2044608 = r2044606 * r2044607;
        double r2044609 = t;
        double r2044610 = a;
        double r2044611 = r2044609 * r2044610;
        double r2044612 = r2044608 - r2044611;
        double r2044613 = r2044603 * r2044612;
        double r2044614 = b;
        double r2044615 = c;
        double r2044616 = r2044615 * r2044607;
        double r2044617 = i;
        double r2044618 = r2044617 * r2044610;
        double r2044619 = r2044616 - r2044618;
        double r2044620 = cbrt(r2044619);
        double r2044621 = r2044620 * r2044620;
        double r2044622 = r2044614 * r2044621;
        double r2044623 = r2044622 * r2044620;
        double r2044624 = r2044613 - r2044623;
        double r2044625 = j;
        double r2044626 = r2044615 * r2044609;
        double r2044627 = r2044617 * r2044606;
        double r2044628 = r2044626 - r2044627;
        double r2044629 = r2044625 * r2044628;
        double r2044630 = r2044624 + r2044629;
        double r2044631 = 3.743837311638164e-169;
        bool r2044632 = r2044603 <= r2044631;
        double r2044633 = 0.0;
        double r2044634 = r2044614 * r2044619;
        double r2044635 = r2044633 - r2044634;
        double r2044636 = r2044635 + r2044629;
        double r2044637 = cbrt(r2044614);
        double r2044638 = r2044637 * r2044637;
        double r2044639 = r2044637 * r2044619;
        double r2044640 = r2044638 * r2044639;
        double r2044641 = r2044613 - r2044640;
        double r2044642 = r2044641 + r2044629;
        double r2044643 = r2044632 ? r2044636 : r2044642;
        double r2044644 = r2044605 ? r2044630 : r2044643;
        return r2044644;
}

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	12.5
Target	16.2
Herbie	12.7

\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -3.0773273627394588e-176

   1. Initial program 10.7

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

   3. Applied add-cube-cbrt 10.9

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

   4. Applied associate-*r* 10.9

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

   if -3.0773273627394588e-176 < x < 3.743837311638164e-169

   1. Initial program 17.9

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

   2. Taylor expanded around 0 17.8

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

   if 3.743837311638164e-169 < x

   1. Initial program 10.1

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

   3. Applied add-cube-cbrt 10.4

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

   4. Applied associate-*l* 10.4

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.0773273627394588 \cdot 10^{-176}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 3.74383731163816427 \cdot 10^{-169}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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

  :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)))))