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

Average Error: 11.7 → 9.2

Time: 26.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -7.642798871419527 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(b \cdot \left(\sqrt[3]{z \cdot c - i \cdot t} \cdot \sqrt[3]{z \cdot c - i \cdot t}\right)\right) \cdot \sqrt[3]{z \cdot c - i \cdot t}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 6.910480137132264 \cdot 10^{+91}:\\ \;\;\;\;\left(c \cdot \left(j \cdot a\right) + \left(i \cdot j\right) \cdot \left(-y\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt{j} \cdot \left(\left(c \cdot a - y \cdot i\right) \cdot \sqrt{j}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r45989015 = x;
        double r45989016 = y;
        double r45989017 = z;
        double r45989018 = r45989016 * r45989017;
        double r45989019 = t;
        double r45989020 = a;
        double r45989021 = r45989019 * r45989020;
        double r45989022 = r45989018 - r45989021;
        double r45989023 = r45989015 * r45989022;
        double r45989024 = b;
        double r45989025 = c;
        double r45989026 = r45989025 * r45989017;
        double r45989027 = i;
        double r45989028 = r45989019 * r45989027;
        double r45989029 = r45989026 - r45989028;
        double r45989030 = r45989024 * r45989029;
        double r45989031 = r45989023 - r45989030;
        double r45989032 = j;
        double r45989033 = r45989025 * r45989020;
        double r45989034 = r45989016 * r45989027;
        double r45989035 = r45989033 - r45989034;
        double r45989036 = r45989032 * r45989035;
        double r45989037 = r45989031 + r45989036;
        return r45989037;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r45989038 = j;
        double r45989039 = -7.642798871419527e+20;
        bool r45989040 = r45989038 <= r45989039;
        double r45989041 = y;
        double r45989042 = z;
        double r45989043 = r45989041 * r45989042;
        double r45989044 = t;
        double r45989045 = a;
        double r45989046 = r45989044 * r45989045;
        double r45989047 = r45989043 - r45989046;
        double r45989048 = x;
        double r45989049 = r45989047 * r45989048;
        double r45989050 = b;
        double r45989051 = c;
        double r45989052 = r45989042 * r45989051;
        double r45989053 = i;
        double r45989054 = r45989053 * r45989044;
        double r45989055 = r45989052 - r45989054;
        double r45989056 = cbrt(r45989055);
        double r45989057 = r45989056 * r45989056;
        double r45989058 = r45989050 * r45989057;
        double r45989059 = r45989058 * r45989056;
        double r45989060 = r45989049 - r45989059;
        double r45989061 = r45989051 * r45989045;
        double r45989062 = r45989041 * r45989053;
        double r45989063 = r45989061 - r45989062;
        double r45989064 = r45989038 * r45989063;
        double r45989065 = r45989060 + r45989064;
        double r45989066 = 6.910480137132264e+91;
        bool r45989067 = r45989038 <= r45989066;
        double r45989068 = r45989038 * r45989045;
        double r45989069 = r45989051 * r45989068;
        double r45989070 = r45989053 * r45989038;
        double r45989071 = -r45989041;
        double r45989072 = r45989070 * r45989071;
        double r45989073 = r45989069 + r45989072;
        double r45989074 = r45989050 * r45989055;
        double r45989075 = r45989049 - r45989074;
        double r45989076 = r45989073 + r45989075;
        double r45989077 = sqrt(r45989038);
        double r45989078 = r45989063 * r45989077;
        double r45989079 = r45989077 * r45989078;
        double r45989080 = r45989075 + r45989079;
        double r45989081 = r45989067 ? r45989076 : r45989080;
        double r45989082 = r45989040 ? r45989065 : r45989081;
        return r45989082;
}

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	11.7
Target	18.4
Herbie	9.2

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if j < -7.642798871419527e+20

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt 7.2

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

   4. Applied associate-*r* 7.2

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

   if -7.642798871419527e+20 < j < 6.910480137132264e+91

   1. Initial program 13.7

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

   3. Applied add-cube-cbrt 13.9

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

   4. Applied associate-*l* 13.9

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

   6. Applied sub-neg 13.9

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

   7. Applied distribute-lft-in 13.9

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

   8. Applied distribute-lft-in 13.9

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

   9. Simplified 13.8

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

   10. Simplified 11.8

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

   12. Applied associate-*l* 10.1

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

   if 6.910480137132264e+91 < j

   1. Initial program 7.0

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

   3. Applied add-sqr-sqrt 7.2

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

   4. Applied associate-*l* 7.2

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -7.642798871419527 \cdot 10^{+20}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(b \cdot \left(\sqrt[3]{z \cdot c - i \cdot t} \cdot \sqrt[3]{z \cdot c - i \cdot t}\right)\right) \cdot \sqrt[3]{z \cdot c - i \cdot t}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 6.910480137132264 \cdot 10^{+91}:\\ \;\;\;\;\left(c \cdot \left(j \cdot a\right) + \left(i \cdot j\right) \cdot \left(-y\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt{j} \cdot \left(\left(c \cdot a - y \cdot i\right) \cdot \sqrt{j}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :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)))))