Average Error: 11.6 → 11.7
Time: 58.7s
Precision: 64
Internal Precision: 128
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.522288223726434 \cdot 10^{+72}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\left(\left(x \cdot y\right) \cdot z + \left(\sqrt[3]{\left(-x\right) \cdot t} \cdot \sqrt[3]{\left(-x\right) \cdot t}\right) \cdot \left(a \cdot \sqrt[3]{\left(-x\right) \cdot t}\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right)\\ \mathbf{elif}\;z \le 1.2475157694933916 \cdot 10^{+188}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a\right) \cdot \left(x \cdot t\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(-a \cdot t\right) + \left(x \cdot y\right) \cdot z\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

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 z < -1.522288223726434e+72

    1. Initial program 19.1

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

    3. Applied sub-neg19.1

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

    4. Applied distribute-lft-in19.1

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

    6. Applied associate-*r*14.1

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

    8. Applied distribute-lft-neg-in14.1

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

    9. Applied associate-*r*13.3

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

    11. Applied add-cube-cbrt13.4

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

    12. Applied associate-*l*13.4

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

    if -1.522288223726434e+72 < z < 1.2475157694933916e+188

    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. Using strategy rm

    6. Applied associate-*r*11.4

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

    8. Applied distribute-lft-neg-in11.4

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

    9. Applied associate-*r*11.5

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

    11. Applied *-commutative11.5

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

    12. Applied associate-*l*10.2

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

    if 1.2475157694933916e+188 < z

    1. Initial program 23.1

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

    3. Applied sub-neg23.1

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

    4. Applied distribute-lft-in23.1

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

    6. Applied associate-*r*16.1

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

    7. Taylor expanded around 0 30.2

      \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + x \cdot \left(-t \cdot a\right)\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.522288223726434 \cdot 10^{+72}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\left(\left(x \cdot y\right) \cdot z + \left(\sqrt[3]{\left(-x\right) \cdot t} \cdot \sqrt[3]{\left(-x\right) \cdot t}\right) \cdot \left(a \cdot \sqrt[3]{\left(-x\right) \cdot t}\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right)\\ \mathbf{elif}\;z \le 1.2475157694933916 \cdot 10^{+188}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a\right) \cdot \left(x \cdot t\right)\right) - b \cdot \left(z \cdot c - a \cdot i\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(-a \cdot t\right) + \left(x \cdot y\right) \cdot z\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 
(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)))))