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

Average Error: 11.9 → 11.2

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.84509602191074263570705140237543983583 \cdot 10^{-139}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right) \cdot j\right) \cdot \sqrt[3]{c \cdot a - i \cdot y}\\ \mathbf{elif}\;x \le 8.34714696233437041626136617555834573359 \cdot 10^{-192}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(z \cdot c - i \cdot t\right) \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r43046094 = x;
        double r43046095 = y;
        double r43046096 = z;
        double r43046097 = r43046095 * r43046096;
        double r43046098 = t;
        double r43046099 = a;
        double r43046100 = r43046098 * r43046099;
        double r43046101 = r43046097 - r43046100;
        double r43046102 = r43046094 * r43046101;
        double r43046103 = b;
        double r43046104 = c;
        double r43046105 = r43046104 * r43046096;
        double r43046106 = i;
        double r43046107 = r43046098 * r43046106;
        double r43046108 = r43046105 - r43046107;
        double r43046109 = r43046103 * r43046108;
        double r43046110 = r43046102 - r43046109;
        double r43046111 = j;
        double r43046112 = r43046104 * r43046099;
        double r43046113 = r43046095 * r43046106;
        double r43046114 = r43046112 - r43046113;
        double r43046115 = r43046111 * r43046114;
        double r43046116 = r43046110 + r43046115;
        return r43046116;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r43046117 = x;
        double r43046118 = -1.8450960219107426e-139;
        bool r43046119 = r43046117 <= r43046118;
        double r43046120 = y;
        double r43046121 = z;
        double r43046122 = r43046120 * r43046121;
        double r43046123 = a;
        double r43046124 = t;
        double r43046125 = r43046123 * r43046124;
        double r43046126 = r43046122 - r43046125;
        double r43046127 = r43046126 * r43046117;
        double r43046128 = b;
        double r43046129 = c;
        double r43046130 = r43046121 * r43046129;
        double r43046131 = i;
        double r43046132 = r43046131 * r43046124;
        double r43046133 = r43046130 - r43046132;
        double r43046134 = r43046128 * r43046133;
        double r43046135 = r43046127 - r43046134;
        double r43046136 = r43046129 * r43046123;
        double r43046137 = r43046131 * r43046120;
        double r43046138 = r43046136 - r43046137;
        double r43046139 = cbrt(r43046138);
        double r43046140 = r43046139 * r43046139;
        double r43046141 = j;
        double r43046142 = r43046140 * r43046141;
        double r43046143 = r43046142 * r43046139;
        double r43046144 = r43046135 + r43046143;
        double r43046145 = 8.34714696233437e-192;
        bool r43046146 = r43046117 <= r43046145;
        double r43046147 = r43046117 * r43046122;
        double r43046148 = r43046117 * r43046124;
        double r43046149 = r43046148 * r43046123;
        double r43046150 = r43046147 - r43046149;
        double r43046151 = cbrt(r43046128);
        double r43046152 = r43046151 * r43046151;
        double r43046153 = r43046133 * r43046151;
        double r43046154 = r43046152 * r43046153;
        double r43046155 = r43046150 - r43046154;
        double r43046156 = r43046141 * r43046138;
        double r43046157 = r43046155 + r43046156;
        double r43046158 = sqrt(r43046117);
        double r43046159 = r43046126 * r43046158;
        double r43046160 = r43046158 * r43046159;
        double r43046161 = r43046160 - r43046134;
        double r43046162 = r43046161 + r43046156;
        double r43046163 = r43046146 ? r43046157 : r43046162;
        double r43046164 = r43046119 ? r43046144 : r43046163;
        return r43046164;
}

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	11.9
Target	20.1
Herbie	11.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1.8450960219107426e-139

   1. Initial program 9.4

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

   3. Applied add-cube-cbrt 9.6

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

   4. Applied associate-*r* 9.6

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

   if -1.8450960219107426e-139 < x < 8.34714696233437e-192

   1. Initial program 17.0

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

   3. Applied add-cube-cbrt 17.3

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

   4. Applied associate-*l* 17.3

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

   5. Taylor expanded around inf 14.2

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

   if 8.34714696233437e-192 < x

   1. Initial program 10.3

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

   3. Applied add-sqr-sqrt 10.4

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

   4. Applied associate-*l* 10.4

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.84509602191074263570705140237543983583 \cdot 10^{-139}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right) \cdot j\right) \cdot \sqrt[3]{c \cdot a - i \cdot y}\\ \mathbf{elif}\;x \le 8.34714696233437041626136617555834573359 \cdot 10^{-192}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(z \cdot c - i \cdot t\right) \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\left(y \cdot z - a \cdot t\right) \cdot \sqrt{x}\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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