Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.6 → 13.1

Time: 10.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.400785168655679605773741836481995189038 \cdot 10^{-141} \lor \neg \left(b \le -5.616902006401816366798818804296717843031 \cdot 10^{-291}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r702941 = x;
        double r702942 = y;
        double r702943 = z;
        double r702944 = r702942 * r702943;
        double r702945 = t;
        double r702946 = a;
        double r702947 = r702945 * r702946;
        double r702948 = r702944 - r702947;
        double r702949 = r702941 * r702948;
        double r702950 = b;
        double r702951 = c;
        double r702952 = r702951 * r702943;
        double r702953 = i;
        double r702954 = r702953 * r702946;
        double r702955 = r702952 - r702954;
        double r702956 = r702950 * r702955;
        double r702957 = r702949 - r702956;
        double r702958 = j;
        double r702959 = r702951 * r702945;
        double r702960 = r702953 * r702942;
        double r702961 = r702959 - r702960;
        double r702962 = r702958 * r702961;
        double r702963 = r702957 + r702962;
        return r702963;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r702964 = b;
        double r702965 = -4.4007851686556796e-141;
        bool r702966 = r702964 <= r702965;
        double r702967 = -5.6169020064018164e-291;
        bool r702968 = r702964 <= r702967;
        double r702969 = !r702968;
        bool r702970 = r702966 || r702969;
        double r702971 = x;
        double r702972 = y;
        double r702973 = z;
        double r702974 = a;
        double r702975 = t;
        double r702976 = r702974 * r702975;
        double r702977 = -r702976;
        double r702978 = fma(r702972, r702973, r702977);
        double r702979 = r702971 * r702978;
        double r702980 = cbrt(r702979);
        double r702981 = r702980 * r702980;
        double r702982 = cbrt(r702971);
        double r702983 = cbrt(r702978);
        double r702984 = r702982 * r702983;
        double r702985 = r702981 * r702984;
        double r702986 = -r702974;
        double r702987 = fma(r702986, r702975, r702976);
        double r702988 = r702971 * r702987;
        double r702989 = r702985 + r702988;
        double r702990 = c;
        double r702991 = r702990 * r702973;
        double r702992 = i;
        double r702993 = r702992 * r702974;
        double r702994 = r702991 - r702993;
        double r702995 = r702964 * r702994;
        double r702996 = r702989 - r702995;
        double r702997 = j;
        double r702998 = r702990 * r702975;
        double r702999 = r702992 * r702972;
        double r703000 = r702998 - r702999;
        double r703001 = r702997 * r703000;
        double r703002 = r702996 + r703001;
        double r703003 = r702979 + r702988;
        double r703004 = 0.0;
        double r703005 = r703003 - r703004;
        double r703006 = r703005 + r703001;
        double r703007 = r702970 ? r703002 : r703006;
        return r703007;
}

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	12.6
Target	16.5
Herbie	13.1

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

Derivation

1. Split input into 2 regimes

2. if b < -4.4007851686556796e-141 or -5.6169020064018164e-291 < b

   1. Initial program 11.6

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

   3. Applied prod-diff 11.6

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

   4. Applied distribute-lft-in 11.6

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

   6. Applied add-cube-cbrt 11.8

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

   8. Applied cbrt-prod 11.8

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

   if -4.4007851686556796e-141 < b < -5.6169020064018164e-291

   1. Initial program 18.3

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

   3. Applied prod-diff 18.3

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

   4. Applied distribute-lft-in 18.3

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

   5. Taylor expanded around 0 20.1

      \[\leadsto \left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.400785168655679605773741836481995189038 \cdot 10^{-141} \lor \neg \left(b \le -5.616902006401816366798818804296717843031 \cdot 10^{-291}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(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 -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-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 -7.633533346031584e-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)))))