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

Average Error: 12.6 → 12.8

Time: 8.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1028939 = x;
        double r1028940 = y;
        double r1028941 = z;
        double r1028942 = r1028940 * r1028941;
        double r1028943 = t;
        double r1028944 = a;
        double r1028945 = r1028943 * r1028944;
        double r1028946 = r1028942 - r1028945;
        double r1028947 = r1028939 * r1028946;
        double r1028948 = b;
        double r1028949 = c;
        double r1028950 = r1028949 * r1028941;
        double r1028951 = i;
        double r1028952 = r1028943 * r1028951;
        double r1028953 = r1028950 - r1028952;
        double r1028954 = r1028948 * r1028953;
        double r1028955 = r1028947 - r1028954;
        double r1028956 = j;
        double r1028957 = r1028949 * r1028944;
        double r1028958 = r1028940 * r1028951;
        double r1028959 = r1028957 - r1028958;
        double r1028960 = r1028956 * r1028959;
        double r1028961 = r1028955 + r1028960;
        return r1028961;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1028962 = x;
        double r1028963 = y;
        double r1028964 = z;
        double r1028965 = r1028963 * r1028964;
        double r1028966 = t;
        double r1028967 = a;
        double r1028968 = r1028966 * r1028967;
        double r1028969 = r1028965 - r1028968;
        double r1028970 = r1028962 * r1028969;
        double r1028971 = cbrt(r1028970);
        double r1028972 = r1028971 * r1028971;
        double r1028973 = cbrt(r1028962);
        double r1028974 = cbrt(r1028969);
        double r1028975 = r1028973 * r1028974;
        double r1028976 = r1028972 * r1028975;
        double r1028977 = b;
        double r1028978 = c;
        double r1028979 = r1028978 * r1028964;
        double r1028980 = i;
        double r1028981 = r1028966 * r1028980;
        double r1028982 = r1028979 - r1028981;
        double r1028983 = r1028977 * r1028982;
        double r1028984 = r1028976 - r1028983;
        double r1028985 = j;
        double r1028986 = r1028978 * r1028967;
        double r1028987 = r1028963 * r1028980;
        double r1028988 = r1028986 - r1028987;
        double r1028989 = r1028985 * r1028988;
        double r1028990 = r1028984 + r1028989;
        return r1028990;
}

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	12.6
Target	20.3
Herbie	12.8

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

Derivation

1. Initial program 12.6

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

3. Applied add-cube-cbrt 12.8

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

5. Applied cbrt-prod 12.8

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

6. Final simplification 12.8

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

Reproduce

herbie shell --seed 2019362 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* 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) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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