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

Average Error: 12.5 → 9.4

Time: 27.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -8.098605839800467588832943275516974834805 \cdot 10^{-10} \lor \neg \left(j \le 1.934400928972068227614321657053659421626 \cdot 10^{59}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}, \mathsf{fma}\left(b, t \cdot i - c \cdot z, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, t \cdot i - c \cdot z, a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r669934 = x;
        double r669935 = y;
        double r669936 = z;
        double r669937 = r669935 * r669936;
        double r669938 = t;
        double r669939 = a;
        double r669940 = r669938 * r669939;
        double r669941 = r669937 - r669940;
        double r669942 = r669934 * r669941;
        double r669943 = b;
        double r669944 = c;
        double r669945 = r669944 * r669936;
        double r669946 = i;
        double r669947 = r669938 * r669946;
        double r669948 = r669945 - r669947;
        double r669949 = r669943 * r669948;
        double r669950 = r669942 - r669949;
        double r669951 = j;
        double r669952 = r669944 * r669939;
        double r669953 = r669935 * r669946;
        double r669954 = r669952 - r669953;
        double r669955 = r669951 * r669954;
        double r669956 = r669950 + r669955;
        return r669956;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r669957 = j;
        double r669958 = -8.098605839800468e-10;
        bool r669959 = r669957 <= r669958;
        double r669960 = 1.9344009289720682e+59;
        bool r669961 = r669957 <= r669960;
        double r669962 = !r669961;
        bool r669963 = r669959 || r669962;
        double r669964 = x;
        double r669965 = y;
        double r669966 = z;
        double r669967 = r669965 * r669966;
        double r669968 = t;
        double r669969 = a;
        double r669970 = r669968 * r669969;
        double r669971 = r669967 - r669970;
        double r669972 = cbrt(r669971);
        double r669973 = r669972 * r669972;
        double r669974 = r669973 * r669972;
        double r669975 = b;
        double r669976 = i;
        double r669977 = r669968 * r669976;
        double r669978 = c;
        double r669979 = r669978 * r669966;
        double r669980 = r669977 - r669979;
        double r669981 = r669978 * r669969;
        double r669982 = r669965 * r669976;
        double r669983 = r669981 - r669982;
        double r669984 = r669957 * r669983;
        double r669985 = fma(r669975, r669980, r669984);
        double r669986 = fma(r669964, r669974, r669985);
        double r669987 = r669957 * r669978;
        double r669988 = r669969 * r669987;
        double r669989 = r669965 * r669957;
        double r669990 = r669976 * r669989;
        double r669991 = -r669990;
        double r669992 = r669988 + r669991;
        double r669993 = fma(r669975, r669980, r669992);
        double r669994 = fma(r669964, r669971, r669993);
        double r669995 = r669963 ? r669986 : r669994;
        return r669995;
}

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	12.5
Target	20.4
Herbie	9.4

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

Derivation

1. Split input into 2 regimes

2. if j < -8.098605839800468e-10 or 1.9344009289720682e+59 < j

   1. Initial program 8.0

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

   2. Simplified 8.0

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

   4. Applied add-cube-cbrt 8.2

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

   if -8.098605839800468e-10 < j < 1.9344009289720682e+59

   1. Initial program 15.1

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

   2. Simplified 15.1

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

   4. Applied add-cube-cbrt 15.3

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

   5. Applied associate-*l* 15.3

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

   7. Applied sub-neg 15.3

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

   8. Applied distribute-lft-in 15.3

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

   9. Applied distribute-lft-in 15.3

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

   10. Simplified 13.0

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

   11. Simplified 10.0

       \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, t \cdot i - c \cdot z, a \cdot \left(j \cdot c\right) + \color{blue}{\left(-i \cdot \left(y \cdot j\right)\right)}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -8.098605839800467588832943275516974834805 \cdot 10^{-10} \lor \neg \left(j \le 1.934400928972068227614321657053659421626 \cdot 10^{59}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}, \mathsf{fma}\left(b, t \cdot i - c \cdot z, j \cdot \left(c \cdot a - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, t \cdot i - c \cdot z, a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))))