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

Average Error: 7.6 → 4.8

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.15371004913535115448520191961472535508 \cdot 10^{201}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a}}, \frac{1}{\frac{\frac{a}{y}}{x}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;a \le 2.040075352769508416644879369622871361138 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a}}, \frac{x}{\frac{a}{y}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \mathsf{fma}\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt{a}}}, -\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{a}}}, \frac{z}{\sqrt[3]{a}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r518956 = x;
        double r518957 = y;
        double r518958 = r518956 * r518957;
        double r518959 = z;
        double r518960 = t;
        double r518961 = r518959 * r518960;
        double r518962 = r518958 - r518961;
        double r518963 = a;
        double r518964 = r518962 / r518963;
        return r518964;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r518965 = a;
        double r518966 = -3.153710049135351e+201;
        bool r518967 = r518965 <= r518966;
        double r518968 = t;
        double r518969 = cbrt(r518965);
        double r518970 = r518969 * r518969;
        double r518971 = r518968 / r518970;
        double r518972 = -r518971;
        double r518973 = z;
        double r518974 = r518973 / r518969;
        double r518975 = 1.0;
        double r518976 = y;
        double r518977 = r518965 / r518976;
        double r518978 = x;
        double r518979 = r518977 / r518978;
        double r518980 = r518975 / r518979;
        double r518981 = fma(r518972, r518974, r518980);
        double r518982 = -r518974;
        double r518983 = r518982 + r518974;
        double r518984 = r518971 * r518983;
        double r518985 = r518981 + r518984;
        double r518986 = 2.0400753527695084e-47;
        bool r518987 = r518965 <= r518986;
        double r518988 = r518975 / r518965;
        double r518989 = r518978 * r518976;
        double r518990 = r518968 * r518973;
        double r518991 = r518989 - r518990;
        double r518992 = r518988 * r518991;
        double r518993 = r518978 / r518977;
        double r518994 = fma(r518972, r518974, r518993);
        double r518995 = cbrt(r518973);
        double r518996 = r518995 * r518995;
        double r518997 = sqrt(r518965);
        double r518998 = cbrt(r518997);
        double r518999 = r518996 / r518998;
        double r519000 = r518995 / r518998;
        double r519001 = -r519000;
        double r519002 = fma(r518999, r519001, r518974);
        double r519003 = r518971 * r519002;
        double r519004 = r518994 + r519003;
        double r519005 = r518987 ? r518992 : r519004;
        double r519006 = r518967 ? r518985 : r519005;
        return r519006;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.6
Target	6.0
Herbie	4.8

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

Derivation

1. Split input into 3 regimes

2. if a < -3.153710049135351e+201

   1. Initial program 15.7

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

   3. Applied div-sub 15.7

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

   4. Simplified 15.7

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

   6. Applied add-cube-cbrt 16.0

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

   7. Applied times-frac 11.5

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

   8. Applied add-sqr-sqrt 28.1

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

   9. Applied prod-diff 28.1

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

   10. Simplified 7.2

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

   11. Simplified 7.2

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

   13. Applied clear-num 7.4

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

   if -3.153710049135351e+201 < a < 2.0400753527695084e-47

   1. Initial program 4.1

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

   3. Applied div-sub 4.1

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

   4. Simplified 4.1

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

   6. Applied *-un-lft-identity 4.1

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

   7. Applied *-un-lft-identity 4.1

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

   8. Applied distribute-lft-out-- 4.1

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

   9. Simplified 4.2

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

   if 2.0400753527695084e-47 < a

   1. Initial program 9.9

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

   3. Applied div-sub 9.9

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

   4. Simplified 9.9

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

   6. Applied add-cube-cbrt 10.3

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

   7. Applied times-frac 7.5

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

   8. Applied add-sqr-sqrt 27.4

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

   9. Applied prod-diff 27.4

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

   10. Simplified 4.7

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

   11. Simplified 4.7

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

   13. Applied add-sqr-sqrt 4.7

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

   14. Applied cbrt-prod 4.7

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

   15. Applied add-cube-cbrt 4.9

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

   16. Applied times-frac 4.9

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

   17. Applied distribute-rgt-neg-in 4.9

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

   18. Applied fma-def 4.9

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

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.15371004913535115448520191961472535508 \cdot 10^{201}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a}}, \frac{1}{\frac{\frac{a}{y}}{x}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{elif}\;a \le 2.040075352769508416644879369622871361138 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a}}, \frac{x}{\frac{a}{y}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \mathsf{fma}\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt{a}}}, -\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt{a}}}, \frac{z}{\sqrt[3]{a}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))