Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 12.4

Time: 14.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -3.06881634392199504 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(i \cdot a - c \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b} + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;j \le 1.143058614838747 \cdot 10^{-175}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - z \cdot \left(b \cdot c\right)\right) + \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - \left(z \cdot b\right) \cdot c\right) + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r582977 = x;
        double r582978 = y;
        double r582979 = z;
        double r582980 = r582978 * r582979;
        double r582981 = t;
        double r582982 = a;
        double r582983 = r582981 * r582982;
        double r582984 = r582980 - r582983;
        double r582985 = r582977 * r582984;
        double r582986 = b;
        double r582987 = c;
        double r582988 = r582987 * r582979;
        double r582989 = i;
        double r582990 = r582989 * r582982;
        double r582991 = r582988 - r582990;
        double r582992 = r582986 * r582991;
        double r582993 = r582985 - r582992;
        double r582994 = j;
        double r582995 = r582987 * r582981;
        double r582996 = r582989 * r582978;
        double r582997 = r582995 - r582996;
        double r582998 = r582994 * r582997;
        double r582999 = r582993 + r582998;
        return r582999;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r583000 = j;
        double r583001 = -3.068816343921995e-18;
        bool r583002 = r583000 <= r583001;
        double r583003 = i;
        double r583004 = a;
        double r583005 = r583003 * r583004;
        double r583006 = c;
        double r583007 = z;
        double r583008 = r583006 * r583007;
        double r583009 = r583005 - r583008;
        double r583010 = b;
        double r583011 = cbrt(r583010);
        double r583012 = r583011 * r583011;
        double r583013 = r583009 * r583012;
        double r583014 = r583013 * r583011;
        double r583015 = t;
        double r583016 = r583006 * r583015;
        double r583017 = y;
        double r583018 = r583003 * r583017;
        double r583019 = r583016 - r583018;
        double r583020 = x;
        double r583021 = r583017 * r583007;
        double r583022 = r583015 * r583004;
        double r583023 = r583021 - r583022;
        double r583024 = r583020 * r583023;
        double r583025 = fma(r583000, r583019, r583024);
        double r583026 = r583014 + r583025;
        double r583027 = 1.143058614838747e-175;
        bool r583028 = r583000 <= r583027;
        double r583029 = r583003 * r583010;
        double r583030 = r583004 * r583029;
        double r583031 = r583010 * r583006;
        double r583032 = r583007 * r583031;
        double r583033 = r583030 - r583032;
        double r583034 = r583004 * r583015;
        double r583035 = -r583034;
        double r583036 = fma(r583017, r583007, r583035);
        double r583037 = r583020 * r583036;
        double r583038 = r583000 * r583017;
        double r583039 = r583003 * r583038;
        double r583040 = r583037 - r583039;
        double r583041 = r583033 + r583040;
        double r583042 = r583007 * r583010;
        double r583043 = r583042 * r583006;
        double r583044 = r583030 - r583043;
        double r583045 = r583044 + r583025;
        double r583046 = r583028 ? r583041 : r583045;
        double r583047 = r583002 ? r583026 : r583046;
        return r583047;
}

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	12.1
Target	16.0
Herbie	12.4

\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if j < -3.068816343921995e-18

   1. Initial program 7.4

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

   2. Simplified 7.4

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

   4. Applied fma-udef 7.4

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

   6. Applied add-cube-cbrt 7.6

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

   7. Applied associate-*r* 7.6

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

   if -3.068816343921995e-18 < j < 1.143058614838747e-175

   1. Initial program 16.2

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

   2. Simplified 16.2

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

   4. Applied fma-udef 16.2

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

   6. Applied add-cube-cbrt 16.5

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

   7. Applied associate-*r* 16.5

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

   8. Taylor expanded around inf 16.9

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

   9. Taylor expanded around inf 16.1

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

   10. Simplified 16.2

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

   if 1.143058614838747e-175 < j

   1. Initial program 9.9

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

   2. Simplified 9.9

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

   4. Applied fma-udef 9.9

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

   6. Applied add-cube-cbrt 10.2

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

   7. Applied associate-*r* 10.2

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

   8. Taylor expanded around inf 11.1

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

   10. Applied associate-*r* 10.7

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -3.06881634392199504 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(i \cdot a - c \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b} + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;j \le 1.143058614838747 \cdot 10^{-175}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - z \cdot \left(b \cdot c\right)\right) + \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - \left(z \cdot b\right) \cdot c\right) + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))))