Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.6 → 12.1

Time: 14.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.640681293042328931603104592550108272228 \cdot 10^{-146} \lor \neg \left(b \le 1.309719638135000977575221737891349424931 \cdot 10^{-119}\right):\\ \;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r586132 = x;
        double r586133 = y;
        double r586134 = z;
        double r586135 = r586133 * r586134;
        double r586136 = t;
        double r586137 = a;
        double r586138 = r586136 * r586137;
        double r586139 = r586135 - r586138;
        double r586140 = r586132 * r586139;
        double r586141 = b;
        double r586142 = c;
        double r586143 = r586142 * r586134;
        double r586144 = i;
        double r586145 = r586144 * r586137;
        double r586146 = r586143 - r586145;
        double r586147 = r586141 * r586146;
        double r586148 = r586140 - r586147;
        double r586149 = j;
        double r586150 = r586142 * r586136;
        double r586151 = r586144 * r586133;
        double r586152 = r586150 - r586151;
        double r586153 = r586149 * r586152;
        double r586154 = r586148 + r586153;
        return r586154;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r586155 = b;
        double r586156 = -1.640681293042329e-146;
        bool r586157 = r586155 <= r586156;
        double r586158 = 1.309719638135001e-119;
        bool r586159 = r586155 <= r586158;
        double r586160 = !r586159;
        bool r586161 = r586157 || r586160;
        double r586162 = j;
        double r586163 = c;
        double r586164 = t;
        double r586165 = r586163 * r586164;
        double r586166 = i;
        double r586167 = y;
        double r586168 = r586166 * r586167;
        double r586169 = r586165 - r586168;
        double r586170 = r586162 * r586169;
        double r586171 = x;
        double r586172 = z;
        double r586173 = r586167 * r586172;
        double r586174 = a;
        double r586175 = r586164 * r586174;
        double r586176 = r586173 - r586175;
        double r586177 = r586171 * r586176;
        double r586178 = r586170 + r586177;
        double r586179 = r586163 * r586172;
        double r586180 = r586166 * r586174;
        double r586181 = r586179 - r586180;
        double r586182 = r586181 * r586155;
        double r586183 = r586178 - r586182;
        double r586184 = r586161 ? r586183 : r586178;
        return r586184;
}

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	11.6
Target	15.3
Herbie	12.1

\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -1.640681293042329e-146

   1. Initial program 9.4

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

   3. Applied add-cube-cbrt 9.6

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

   if -1.640681293042329e-146 < b < 1.309719638135001e-119

   1. Initial program 16.8

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

   2. Taylor expanded around 0 18.2

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if 1.309719638135001e-119 < b

   1. Initial program 8.1

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

   3. Applied add-sqr-sqrt 8.2

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

   4. Applied associate-*l* 8.2

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

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.640681293042328931603104592550108272228 \cdot 10^{-146} \lor \neg \left(b \le 1.309719638135000977575221737891349424931 \cdot 10^{-119}\right):\\ \;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.1209789191959122e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.7125538182184851e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.63353334603158369e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

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