Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 10.7

Time: 8.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -1.76023307603761202409938293253843061099 \cdot 10^{58}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(\sqrt[3]{x \cdot \left(-t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(-t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(-t \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;j \le 1.516551054990740022970306969909426253394 \cdot 10^{61}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;j \le 2.85640851406350578387766548712466692209 \cdot 10^{304}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + j \cdot \left(-i \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r586968 = x;
        double r586969 = y;
        double r586970 = z;
        double r586971 = r586969 * r586970;
        double r586972 = t;
        double r586973 = a;
        double r586974 = r586972 * r586973;
        double r586975 = r586971 - r586974;
        double r586976 = r586968 * r586975;
        double r586977 = b;
        double r586978 = c;
        double r586979 = r586978 * r586970;
        double r586980 = i;
        double r586981 = r586980 * r586973;
        double r586982 = r586979 - r586981;
        double r586983 = r586977 * r586982;
        double r586984 = r586976 - r586983;
        double r586985 = j;
        double r586986 = r586978 * r586972;
        double r586987 = r586980 * r586969;
        double r586988 = r586986 - r586987;
        double r586989 = r586985 * r586988;
        double r586990 = r586984 + r586989;
        return r586990;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r586991 = j;
        double r586992 = -1.760233076037612e+58;
        bool r586993 = r586991 <= r586992;
        double r586994 = x;
        double r586995 = y;
        double r586996 = z;
        double r586997 = r586995 * r586996;
        double r586998 = r586994 * r586997;
        double r586999 = t;
        double r587000 = a;
        double r587001 = r586999 * r587000;
        double r587002 = -r587001;
        double r587003 = r586994 * r587002;
        double r587004 = cbrt(r587003);
        double r587005 = r587004 * r587004;
        double r587006 = r587005 * r587004;
        double r587007 = r586998 + r587006;
        double r587008 = b;
        double r587009 = c;
        double r587010 = r587009 * r586996;
        double r587011 = i;
        double r587012 = r587011 * r587000;
        double r587013 = r587010 - r587012;
        double r587014 = r587008 * r587013;
        double r587015 = r587007 - r587014;
        double r587016 = r587009 * r586999;
        double r587017 = r586991 * r587016;
        double r587018 = r587011 * r586995;
        double r587019 = -r587018;
        double r587020 = r586991 * r587019;
        double r587021 = r587017 + r587020;
        double r587022 = r587015 + r587021;
        double r587023 = 1.51655105499074e+61;
        bool r587024 = r586991 <= r587023;
        double r587025 = r586998 + r587003;
        double r587026 = r587025 - r587014;
        double r587027 = r586991 * r587009;
        double r587028 = r587027 * r586999;
        double r587029 = r587028 + r587020;
        double r587030 = r587026 + r587029;
        double r587031 = 2.856408514063506e+304;
        bool r587032 = r586991 <= r587031;
        double r587033 = r586994 * r586995;
        double r587034 = r587033 * r586996;
        double r587035 = r587034 + r587003;
        double r587036 = r587035 - r587014;
        double r587037 = r587036 + r587021;
        double r587038 = r587032 ? r587037 : r587030;
        double r587039 = r587024 ? r587030 : r587038;
        double r587040 = r586993 ? r587022 : r587039;
        return r587040;
}

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	12.3
Target	16.2
Herbie	10.7

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

Derivation

1. Split input into 3 regimes

2. if j < -1.760233076037612e+58

   1. Initial program 7.8

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

   3. Applied sub-neg 7.8

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

   4. Applied distribute-lft-in 7.8

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

   6. Applied sub-neg 7.8

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

   7. Applied distribute-lft-in 7.8

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

   9. Applied add-cube-cbrt 7.9

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

   if -1.760233076037612e+58 < j < 1.51655105499074e+61 or 2.856408514063506e+304 < j

   1. Initial program 14.2

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

   3. Applied sub-neg 14.2

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

   4. Applied distribute-lft-in 14.2

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

   6. Applied sub-neg 14.2

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

   7. Applied distribute-lft-in 14.2

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

   9. Applied associate-*r* 12.1

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

   if 1.51655105499074e+61 < j < 2.856408514063506e+304

   1. Initial program 6.5

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

   3. Applied sub-neg 6.5

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

   4. Applied distribute-lft-in 6.5

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

   6. Applied sub-neg 6.5

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

   7. Applied distribute-lft-in 6.5

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

   9. Applied associate-*r* 5.9

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.76023307603761202409938293253843061099 \cdot 10^{58}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(\sqrt[3]{x \cdot \left(-t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(-t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(-t \cdot a\right)}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;j \le 1.516551054990740022970306969909426253394 \cdot 10^{61}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{elif}\;j \le 2.85640851406350578387766548712466692209 \cdot 10^{304}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(j \cdot c\right) \cdot t + j \cdot \left(-i \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))))