Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 12.3 → 12.9

Time: 30.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.67781928192120638243806754718986205394 \cdot 10^{-82}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\\ \mathbf{elif}\;x \le 1.061840208015286687867460002036346734688 \cdot 10^{-159}:\\ \;\;\;\;\left(-b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r55897718 = x;
        double r55897719 = y;
        double r55897720 = z;
        double r55897721 = r55897719 * r55897720;
        double r55897722 = t;
        double r55897723 = a;
        double r55897724 = r55897722 * r55897723;
        double r55897725 = r55897721 - r55897724;
        double r55897726 = r55897718 * r55897725;
        double r55897727 = b;
        double r55897728 = c;
        double r55897729 = r55897728 * r55897720;
        double r55897730 = i;
        double r55897731 = r55897722 * r55897730;
        double r55897732 = r55897729 - r55897731;
        double r55897733 = r55897727 * r55897732;
        double r55897734 = r55897726 - r55897733;
        double r55897735 = j;
        double r55897736 = r55897728 * r55897723;
        double r55897737 = r55897719 * r55897730;
        double r55897738 = r55897736 - r55897737;
        double r55897739 = r55897735 * r55897738;
        double r55897740 = r55897734 + r55897739;
        return r55897740;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r55897741 = x;
        double r55897742 = -2.6778192819212064e-82;
        bool r55897743 = r55897741 <= r55897742;
        double r55897744 = y;
        double r55897745 = z;
        double r55897746 = r55897744 * r55897745;
        double r55897747 = t;
        double r55897748 = a;
        double r55897749 = r55897747 * r55897748;
        double r55897750 = r55897746 - r55897749;
        double r55897751 = r55897741 * r55897750;
        double r55897752 = b;
        double r55897753 = c;
        double r55897754 = r55897753 * r55897745;
        double r55897755 = i;
        double r55897756 = r55897747 * r55897755;
        double r55897757 = r55897754 - r55897756;
        double r55897758 = r55897752 * r55897757;
        double r55897759 = r55897751 - r55897758;
        double r55897760 = j;
        double r55897761 = cbrt(r55897760);
        double r55897762 = r55897753 * r55897748;
        double r55897763 = r55897744 * r55897755;
        double r55897764 = r55897762 - r55897763;
        double r55897765 = cbrt(r55897764);
        double r55897766 = r55897761 * r55897765;
        double r55897767 = r55897760 * r55897764;
        double r55897768 = cbrt(r55897767);
        double r55897769 = r55897766 * r55897768;
        double r55897770 = r55897769 * r55897766;
        double r55897771 = r55897759 + r55897770;
        double r55897772 = 1.0618402080152867e-159;
        bool r55897773 = r55897741 <= r55897772;
        double r55897774 = -r55897758;
        double r55897775 = r55897774 + r55897767;
        double r55897776 = sqrt(r55897741);
        double r55897777 = r55897776 * r55897750;
        double r55897778 = r55897776 * r55897777;
        double r55897779 = r55897778 - r55897758;
        double r55897780 = r55897779 + r55897767;
        double r55897781 = r55897773 ? r55897775 : r55897780;
        double r55897782 = r55897743 ? r55897771 : r55897781;
        return r55897782;
}

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	12.3
Target	20.2
Herbie	12.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -2.6778192819212064e-82

   1. Initial program 8.5

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

   3. Applied add-cube-cbrt 8.8

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

   5. Applied cbrt-prod 8.7

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

   7. Applied cbrt-prod 8.8

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

   if -2.6778192819212064e-82 < x < 1.0618402080152867e-159

   1. Initial program 17.2

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

   2. Taylor expanded around 0 18.6

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

   if 1.0618402080152867e-159 < x

   1. Initial program 9.9

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

   3. Applied add-sqr-sqrt 10.0

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

   4. Applied associate-*l* 10.0

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.67781928192120638243806754718986205394 \cdot 10^{-82}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\\ \mathbf{elif}\;x \le 1.061840208015286687867460002036346734688 \cdot 10^{-159}:\\ \;\;\;\;\left(-b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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