Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 12.0

Time: 31.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 6.505647406761650844782775648491058656638 \cdot 10^{-159}:\\ \;\;\;\;\left(\left(\left(z \cdot y\right) \cdot x + \left(t \cdot x\right) \cdot \left(-a\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(z \cdot y - t \cdot a\right) \cdot \sqrt{x}\right) \cdot \sqrt{x} - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r26900040 = x;
        double r26900041 = y;
        double r26900042 = z;
        double r26900043 = r26900041 * r26900042;
        double r26900044 = t;
        double r26900045 = a;
        double r26900046 = r26900044 * r26900045;
        double r26900047 = r26900043 - r26900046;
        double r26900048 = r26900040 * r26900047;
        double r26900049 = b;
        double r26900050 = c;
        double r26900051 = r26900050 * r26900042;
        double r26900052 = i;
        double r26900053 = r26900052 * r26900045;
        double r26900054 = r26900051 - r26900053;
        double r26900055 = r26900049 * r26900054;
        double r26900056 = r26900048 - r26900055;
        double r26900057 = j;
        double r26900058 = r26900050 * r26900044;
        double r26900059 = r26900052 * r26900041;
        double r26900060 = r26900058 - r26900059;
        double r26900061 = r26900057 * r26900060;
        double r26900062 = r26900056 + r26900061;
        return r26900062;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r26900063 = x;
        double r26900064 = 6.505647406761651e-159;
        bool r26900065 = r26900063 <= r26900064;
        double r26900066 = z;
        double r26900067 = y;
        double r26900068 = r26900066 * r26900067;
        double r26900069 = r26900068 * r26900063;
        double r26900070 = t;
        double r26900071 = r26900070 * r26900063;
        double r26900072 = a;
        double r26900073 = -r26900072;
        double r26900074 = r26900071 * r26900073;
        double r26900075 = r26900069 + r26900074;
        double r26900076 = c;
        double r26900077 = r26900076 * r26900066;
        double r26900078 = i;
        double r26900079 = r26900078 * r26900072;
        double r26900080 = r26900077 - r26900079;
        double r26900081 = b;
        double r26900082 = r26900080 * r26900081;
        double r26900083 = r26900075 - r26900082;
        double r26900084 = j;
        double r26900085 = r26900076 * r26900070;
        double r26900086 = r26900078 * r26900067;
        double r26900087 = r26900085 - r26900086;
        double r26900088 = r26900084 * r26900087;
        double r26900089 = r26900083 + r26900088;
        double r26900090 = r26900070 * r26900072;
        double r26900091 = r26900068 - r26900090;
        double r26900092 = sqrt(r26900063);
        double r26900093 = r26900091 * r26900092;
        double r26900094 = r26900093 * r26900092;
        double r26900095 = r26900094 - r26900082;
        double r26900096 = r26900095 + r26900088;
        double r26900097 = r26900065 ? r26900089 : r26900096;
        return r26900097;
}

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	12.2
Target	16.3
Herbie	12.0

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

Derivation

1. Split input into 2 regimes

2. if x < 6.505647406761651e-159

   1. Initial program 13.3

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

   3. Applied add-cube-cbrt 13.6

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

   4. Applied associate-*l* 13.6

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

   6. Applied sub-neg 13.6

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

   7. Applied distribute-lft-in 13.6

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

   8. Applied distribute-lft-in 13.6

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

   9. Simplified 13.5

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

   10. Simplified 13.0

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

   if 6.505647406761651e-159 < x

   1. Initial program 10.1

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

   3. Applied add-sqr-sqrt 10.2

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

   4. Applied associate-*l* 10.3

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6.505647406761650844782775648491058656638 \cdot 10^{-159}:\\ \;\;\;\;\left(\left(\left(z \cdot y\right) \cdot x + \left(t \cdot x\right) \cdot \left(-a\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(z \cdot y - t \cdot a\right) \cdot \sqrt{x}\right) \cdot \sqrt{x} - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :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) 2.0) (pow (* i y) 2.0))) (+ (* 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) 2.0) (pow (* i y) 2.0))) (+ (* 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)))))