Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.9 → 12.0

Time: 23.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r28715185 = x;
        double r28715186 = y;
        double r28715187 = z;
        double r28715188 = r28715186 * r28715187;
        double r28715189 = t;
        double r28715190 = a;
        double r28715191 = r28715189 * r28715190;
        double r28715192 = r28715188 - r28715191;
        double r28715193 = r28715185 * r28715192;
        double r28715194 = b;
        double r28715195 = c;
        double r28715196 = r28715195 * r28715187;
        double r28715197 = i;
        double r28715198 = r28715197 * r28715190;
        double r28715199 = r28715196 - r28715198;
        double r28715200 = r28715194 * r28715199;
        double r28715201 = r28715193 - r28715200;
        double r28715202 = j;
        double r28715203 = r28715195 * r28715189;
        double r28715204 = r28715197 * r28715186;
        double r28715205 = r28715203 - r28715204;
        double r28715206 = r28715202 * r28715205;
        double r28715207 = r28715201 + r28715206;
        return r28715207;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r28715208 = t;
        double r28715209 = c;
        double r28715210 = r28715208 * r28715209;
        double r28715211 = i;
        double r28715212 = y;
        double r28715213 = r28715211 * r28715212;
        double r28715214 = r28715210 - r28715213;
        double r28715215 = j;
        double r28715216 = r28715214 * r28715215;
        double r28715217 = z;
        double r28715218 = r28715217 * r28715212;
        double r28715219 = a;
        double r28715220 = r28715208 * r28715219;
        double r28715221 = r28715218 - r28715220;
        double r28715222 = x;
        double r28715223 = r28715221 * r28715222;
        double r28715224 = b;
        double r28715225 = -r28715224;
        double r28715226 = r28715219 * r28715225;
        double r28715227 = r28715226 * r28715211;
        double r28715228 = cbrt(r28715217);
        double r28715229 = r28715224 * r28715209;
        double r28715230 = r28715228 * r28715229;
        double r28715231 = r28715228 * r28715228;
        double r28715232 = r28715230 * r28715231;
        double r28715233 = r28715227 + r28715232;
        double r28715234 = r28715223 - r28715233;
        double r28715235 = r28715216 + r28715234;
        return r28715235;
}

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	11.9
Target	15.3
Herbie	12.0

\[\begin{array}{l} \mathbf{if}\;t \lt -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Initial program 11.9

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

3. Applied sub-neg 11.9

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

4. Applied distribute-rgt-in 11.9

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

6. Applied distribute-lft-neg-in 11.9

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

7. Applied associate-*l* 12.0

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

8. Taylor expanded around inf 11.9

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

10. Applied add-cube-cbrt 12.0

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

11. Applied associate-*l* 12.0

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

12. Final simplification 12.0

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

Reproduce

herbie shell --seed 2019163 
(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) (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)))))