Average Error: 11.9 → 9.8
Time: 2.7m
Precision: 64
Internal Precision: 384
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.9038404700677242 \cdot 10^{-150}:\\ \;\;\;\;\left(a \cdot \left(b \cdot i\right) + \left(t \cdot \left(j \cdot c - x \cdot a\right) - \left(b \cdot c - y \cdot x\right) \cdot z\right)\right) + j \cdot \left(-i \cdot y\right)\\ \mathbf{if}\;t \le 2.4637080062712413 \cdot 10^{-270}:\\ \;\;\;\;\left(a \cdot \left(b \cdot i - x \cdot t\right) - \left(\left(c \cdot b - y \cdot x\right) \cdot z - c \cdot \left(t \cdot j\right)\right)\right) + \left(j \cdot \left(-i\right)\right) \cdot y\\ \mathbf{if}\;t \le 7.45776387069902 \cdot 10^{+137}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(b \cdot i\right) + \left(t \cdot \left(j \cdot c - x \cdot a\right) - \left(b \cdot c - y \cdot x\right) \cdot z\right)\right) + j \cdot \left(-i \cdot y\right)\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if t < -1.9038404700677242e-150 or 7.45776387069902e+137 < t

    1. Initial program 14.8

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

    3. Applied sub-neg14.8

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

    4. Applied distribute-lft-in14.8

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

    5. Applied associate--l+14.8

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

    6. Applied simplify15.2

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

    8. Applied sub-neg15.2

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

    9. Applied distribute-lft-in15.2

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

    10. Applied associate-+r+15.2

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

    11. Applied simplify14.8

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

    13. Applied sub-neg14.8

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

    14. Applied distribute-lft-in14.8

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

    15. Applied associate--l+14.8

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

    16. Applied simplify8.9

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

    if -1.9038404700677242e-150 < t < 2.4637080062712413e-270

    1. Initial program 9.5

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

    3. Applied sub-neg9.5

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

    4. Applied distribute-lft-in9.5

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

    5. Applied associate--l+9.5

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

    6. Applied simplify9.4

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

    8. Applied sub-neg9.4

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

    9. Applied distribute-lft-in9.4

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

    10. Applied associate-+r+9.4

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

    11. Applied simplify10.4

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

    13. Applied distribute-lft-neg-in10.4

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

    14. Applied associate-*r*11.4

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

    if 2.4637080062712413e-270 < t < 7.45776387069902e+137

    1. Initial program 9.6

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

    3. Applied add-cube-cbrt9.9

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

    4. Applied associate-*r*9.9

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))