Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 9.8

Time: 13.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.83346635036375763 \cdot 10^{180}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;b \le 0.0155641473725553429:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(\left(-i\right) \cdot b\right) \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r430642 = x;
        double r430643 = y;
        double r430644 = z;
        double r430645 = r430643 * r430644;
        double r430646 = t;
        double r430647 = a;
        double r430648 = r430646 * r430647;
        double r430649 = r430645 - r430648;
        double r430650 = r430642 * r430649;
        double r430651 = b;
        double r430652 = c;
        double r430653 = r430652 * r430644;
        double r430654 = i;
        double r430655 = r430654 * r430647;
        double r430656 = r430653 - r430655;
        double r430657 = r430651 * r430656;
        double r430658 = r430650 - r430657;
        double r430659 = j;
        double r430660 = r430652 * r430646;
        double r430661 = r430654 * r430643;
        double r430662 = r430660 - r430661;
        double r430663 = r430659 * r430662;
        double r430664 = r430658 + r430663;
        return r430664;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r430665 = b;
        double r430666 = -6.833466350363758e+180;
        bool r430667 = r430665 <= r430666;
        double r430668 = x;
        double r430669 = y;
        double r430670 = z;
        double r430671 = r430669 * r430670;
        double r430672 = t;
        double r430673 = a;
        double r430674 = r430672 * r430673;
        double r430675 = r430671 - r430674;
        double r430676 = r430668 * r430675;
        double r430677 = c;
        double r430678 = r430677 * r430670;
        double r430679 = i;
        double r430680 = r430679 * r430673;
        double r430681 = r430678 - r430680;
        double r430682 = r430665 * r430681;
        double r430683 = r430676 - r430682;
        double r430684 = j;
        double r430685 = cbrt(r430684);
        double r430686 = r430685 * r430685;
        double r430687 = r430677 * r430672;
        double r430688 = r430679 * r430669;
        double r430689 = r430687 - r430688;
        double r430690 = r430685 * r430689;
        double r430691 = r430686 * r430690;
        double r430692 = r430683 + r430691;
        double r430693 = 0.015564147372555343;
        bool r430694 = r430665 <= r430693;
        double r430695 = r430670 * r430665;
        double r430696 = r430695 * r430677;
        double r430697 = -r430679;
        double r430698 = r430697 * r430665;
        double r430699 = r430698 * r430673;
        double r430700 = r430696 + r430699;
        double r430701 = r430676 - r430700;
        double r430702 = r430687 * r430684;
        double r430703 = r430684 * r430669;
        double r430704 = r430679 * r430703;
        double r430705 = -r430704;
        double r430706 = r430702 + r430705;
        double r430707 = r430701 + r430706;
        double r430708 = r430679 * r430684;
        double r430709 = r430708 * r430669;
        double r430710 = -r430709;
        double r430711 = r430702 + r430710;
        double r430712 = r430683 + r430711;
        double r430713 = r430694 ? r430707 : r430712;
        double r430714 = r430667 ? r430692 : r430713;
        return r430714;
}

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	16.0
Herbie	9.8

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

Derivation

1. Split input into 3 regimes

2. if b < -6.833466350363758e+180

   1. Initial program 5.5

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

   3. Applied add-cube-cbrt 5.7

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

   4. Applied associate-*l* 5.7

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

   if -6.833466350363758e+180 < b < 0.015564147372555343

   1. Initial program 14.0

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

   3. Applied sub-neg 14.0

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

   4. Applied distribute-lft-in 14.0

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

   5. Simplified 14.0

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

   6. Simplified 14.0

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

   8. Applied sub-neg 14.0

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

   9. Applied distribute-lft-in 14.0

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

   10. Simplified 12.9

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

   12. Applied distribute-lft-neg-in 12.9

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

   13. Applied associate-*r* 11.0

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

   14. Simplified 11.0

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

   16. Applied associate-*r* 10.8

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

   if 0.015564147372555343 < b

   1. Initial program 6.8

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

   3. Applied sub-neg 6.8

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

   4. Applied distribute-lft-in 6.8

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

   5. Simplified 6.8

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

   6. Simplified 7.9

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

   8. Applied associate-*r* 7.2

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.83346635036375763 \cdot 10^{180}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;b \le 0.0155641473725553429:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(\left(-i\right) \cdot b\right) \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))