Average Error: 11.3 → 9.1
Time: 34.9s
Precision: 64
Internal Precision: 576
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -1112321566896.243 \lor \neg \left(j \le 1.2362946222159552 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(a \cdot i\right) \cdot \left(-b\right) + \left(c \cdot b\right) \cdot z\right)\right) + j \cdot \left(t \cdot c - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(z \cdot c\right) \cdot b + \left(a \cdot i\right) \cdot \left(-b\right)\right)\right) + \left(\left(y \cdot j\right) \cdot \left(-i\right) + t \cdot \left(c \cdot j\right)\right)\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if j < -1112321566896.243 or 1.2362946222159552e-39 < j

    1. Initial program 7.9

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

    3. Applied sub-neg7.9

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

    4. Applied distribute-lft-in7.9

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

    5. Taylor expanded around -inf 8.5

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

    if -1112321566896.243 < j < 1.2362946222159552e-39

    1. Initial program 13.9

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

    3. Applied sub-neg13.9

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

    4. Applied distribute-lft-in13.9

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

    6. Applied sub-neg13.9

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

    7. Applied distribute-rgt-in13.9

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

    8. Taylor expanded around -inf 11.7

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

    9. Taylor expanded around inf 9.6

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-i \cdot a\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) + \color{blue}{-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)}\right)\]

    10. Simplified9.6

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1112321566896.243 \lor \neg \left(j \le 1.2362946222159552 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(a \cdot i\right) \cdot \left(-b\right) + \left(c \cdot b\right) \cdot z\right)\right) + j \cdot \left(t \cdot c - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(z \cdot c\right) \cdot b + \left(a \cdot i\right) \cdot \left(-b\right)\right)\right) + \left(\left(y \cdot j\right) \cdot \left(-i\right) + t \cdot \left(c \cdot j\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 34.9s)Debug logProfile

herbie shell --seed 2018254 
(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)))))