Data.Colour.Matrix:inverse from colour-2.3.3, B

Average Error: 7.2 → 2.5

Time: 6.4s

Precision: 64

Math

\[\frac{x \cdot y - z \cdot t}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.969844255265401280863746774205918097988 \cdot 10^{306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 6.641618666330047187114329010525050890954 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 1.036441237581747058313870687243839130666 \cdot 10^{299}:\\ \;\;\;\;\sqrt{x \cdot y - z \cdot t} \cdot \frac{\sqrt{x \cdot y - z \cdot t}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r861819 = x;
        double r861820 = y;
        double r861821 = r861819 * r861820;
        double r861822 = z;
        double r861823 = t;
        double r861824 = r861822 * r861823;
        double r861825 = r861821 - r861824;
        double r861826 = a;
        double r861827 = r861825 / r861826;
        return r861827;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r861828 = x;
        double r861829 = y;
        double r861830 = r861828 * r861829;
        double r861831 = z;
        double r861832 = t;
        double r861833 = r861831 * r861832;
        double r861834 = r861830 - r861833;
        double r861835 = -1.9698442552654013e+306;
        bool r861836 = r861834 <= r861835;
        double r861837 = a;
        double r861838 = cbrt(r861837);
        double r861839 = r861838 * r861838;
        double r861840 = r861828 / r861839;
        double r861841 = r861829 / r861838;
        double r861842 = r861832 / r861837;
        double r861843 = r861831 * r861842;
        double r861844 = -r861843;
        double r861845 = fma(r861840, r861841, r861844);
        double r861846 = r861832 / r861839;
        double r861847 = r861831 / r861838;
        double r861848 = -r861847;
        double r861849 = r861848 + r861847;
        double r861850 = r861846 * r861849;
        double r861851 = r861845 + r861850;
        double r861852 = 6.641618666330047e-262;
        bool r861853 = r861834 <= r861852;
        double r861854 = r861837 / r861829;
        double r861855 = r861828 / r861854;
        double r861856 = r861832 * r861831;
        double r861857 = r861856 / r861837;
        double r861858 = r861855 - r861857;
        double r861859 = 1.036441237581747e+299;
        bool r861860 = r861834 <= r861859;
        double r861861 = sqrt(r861834);
        double r861862 = r861861 / r861837;
        double r861863 = r861861 * r861862;
        double r861864 = r861860 ? r861863 : r861851;
        double r861865 = r861853 ? r861858 : r861864;
        double r861866 = r861836 ? r861851 : r861865;
        return r861866;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.2
Target	5.7
Herbie	2.5

\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548224247694913169778644284 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371209578784129518242708809 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z t)) < -1.9698442552654013e+306 or 1.036441237581747e+299 < (- (* x y) (* z t))

   1. Initial program 60.9

      \[\frac{x \cdot y - z \cdot t}{a}\]
   2. Using strategy rm

   3. Applied div-sub 60.9

      \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}}\]

   4. Simplified 60.9

      \[\leadsto \frac{x \cdot y}{a} - \color{blue}{\frac{t \cdot z}{a}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 60.9

      \[\leadsto \frac{x \cdot y}{a} - \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   7. Applied times-frac 33.2

      \[\leadsto \frac{x \cdot y}{a} - \color{blue}{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\]

   8. Applied add-cube-cbrt 33.2

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]

   9. Applied times-frac 1.3

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]

   10. Applied prod-diff 1.3

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \mathsf{fma}\left(-\frac{z}{\sqrt[3]{a}}, \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}\]

   11. Simplified 1.3

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \color{blue}{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)}\]
   12. Using strategy rm

   13. Applied div-inv 1.3

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\color{blue}{\left(z \cdot \frac{1}{\sqrt[3]{a}}\right)} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\]

   14. Applied associate-*l* 1.3

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\color{blue}{z \cdot \left(\frac{1}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\]

   15. Simplified 0.8

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \color{blue}{\frac{t}{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\]

   if -1.9698442552654013e+306 < (- (* x y) (* z t)) < 6.641618666330047e-262

   1. Initial program 1.1

      \[\frac{x \cdot y - z \cdot t}{a}\]
   2. Using strategy rm

   3. Applied div-sub 1.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}}\]

   4. Simplified 1.1

      \[\leadsto \frac{x \cdot y}{a} - \color{blue}{\frac{t \cdot z}{a}}\]
   5. Using strategy rm

   6. Applied associate-/l* 4.7

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{t \cdot z}{a}\]

   if 6.641618666330047e-262 < (- (* x y) (* z t)) < 1.036441237581747e+299

   1. Initial program 0.3

      \[\frac{x \cdot y - z \cdot t}{a}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.3

      \[\leadsto \frac{x \cdot y - z \cdot t}{\color{blue}{1 \cdot a}}\]

   4. Applied add-sqr-sqrt 0.6

      \[\leadsto \frac{\color{blue}{\sqrt{x \cdot y - z \cdot t} \cdot \sqrt{x \cdot y - z \cdot t}}}{1 \cdot a}\]

   5. Applied times-frac 0.6

      \[\leadsto \color{blue}{\frac{\sqrt{x \cdot y - z \cdot t}}{1} \cdot \frac{\sqrt{x \cdot y - z \cdot t}}{a}}\]

   6. Simplified 0.6

      \[\leadsto \color{blue}{\sqrt{x \cdot y - z \cdot t}} \cdot \frac{\sqrt{x \cdot y - z \cdot t}}{a}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.969844255265401280863746774205918097988 \cdot 10^{306}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 6.641618666330047187114329010525050890954 \cdot 10^{-262}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 1.036441237581747058313870687243839130666 \cdot 10^{299}:\\ \;\;\;\;\sqrt{x \cdot y - z \cdot t} \cdot \frac{\sqrt{x \cdot y - z \cdot t}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))