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

Average Error: 12.5 → 9.2

Time: 26.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.557156540206759491060099266665758475148 \cdot 10^{77} \lor \neg \left(b \le 6.288387807050133384086894370081620679201 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r547020 = x;
        double r547021 = y;
        double r547022 = z;
        double r547023 = r547021 * r547022;
        double r547024 = t;
        double r547025 = a;
        double r547026 = r547024 * r547025;
        double r547027 = r547023 - r547026;
        double r547028 = r547020 * r547027;
        double r547029 = b;
        double r547030 = c;
        double r547031 = r547030 * r547022;
        double r547032 = i;
        double r547033 = r547024 * r547032;
        double r547034 = r547031 - r547033;
        double r547035 = r547029 * r547034;
        double r547036 = r547028 - r547035;
        double r547037 = j;
        double r547038 = r547030 * r547025;
        double r547039 = r547021 * r547032;
        double r547040 = r547038 - r547039;
        double r547041 = r547037 * r547040;
        double r547042 = r547036 + r547041;
        return r547042;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r547043 = b;
        double r547044 = -2.5571565402067595e+77;
        bool r547045 = r547043 <= r547044;
        double r547046 = 6.288387807050133e-14;
        bool r547047 = r547043 <= r547046;
        double r547048 = !r547047;
        bool r547049 = r547045 || r547048;
        double r547050 = x;
        double r547051 = y;
        double r547052 = z;
        double r547053 = r547051 * r547052;
        double r547054 = t;
        double r547055 = a;
        double r547056 = r547054 * r547055;
        double r547057 = r547053 - r547056;
        double r547058 = r547050 * r547057;
        double r547059 = c;
        double r547060 = r547059 * r547052;
        double r547061 = i;
        double r547062 = r547054 * r547061;
        double r547063 = r547060 - r547062;
        double r547064 = r547043 * r547063;
        double r547065 = r547058 - r547064;
        double r547066 = j;
        double r547067 = r547059 * r547055;
        double r547068 = r547051 * r547061;
        double r547069 = r547067 - r547068;
        double r547070 = r547066 * r547069;
        double r547071 = cbrt(r547070);
        double r547072 = r547071 * r547071;
        double r547073 = r547072 * r547071;
        double r547074 = r547065 + r547073;
        double r547075 = r547043 * r547059;
        double r547076 = r547052 * r547075;
        double r547077 = r547061 * r547043;
        double r547078 = r547054 * r547077;
        double r547079 = -r547078;
        double r547080 = r547076 + r547079;
        double r547081 = r547058 - r547080;
        double r547082 = r547070 + r547081;
        double r547083 = r547049 ? r547074 : r547082;
        return r547083;
}

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	9.2

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

Derivation

1. Split input into 2 regimes

2. if b < -2.5571565402067595e+77 or 6.288387807050133e-14 < b

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.6

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

   if -2.5571565402067595e+77 < b < 6.288387807050133e-14

   1. Initial program 15.3

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

   3. Applied sub-neg 15.3

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

   4. Applied distribute-lft-in 15.3

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

   5. Simplified 12.8

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

   7. Applied pow1 12.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \color{blue}{{\left(-t \cdot i\right)}^{1}}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   8. Applied pow1 12.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{{b}^{1}} \cdot {\left(-t \cdot i\right)}^{1}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   9. Applied pow-prod-down 12.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{{\left(b \cdot \left(-t \cdot i\right)\right)}^{1}}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   10. Simplified 10.1

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + {\color{blue}{\left(-t \cdot \left(i \cdot b\right)\right)}}^{1}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.557156540206759491060099266665758475148 \cdot 10^{77} \lor \neg \left(b \le 6.288387807050133384086894370081620679201 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))