Average Error: 11.9 → 9.8
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.4842475539533175 \cdot 10^{+141}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(b \cdot i - t \cdot x\right) \cdot a - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;a \le 7.026026790611426 \cdot 10^{+132}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(i \cdot \left(\left(-y\right) \cdot j\right) + \left(c \cdot j\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(b \cdot i - t \cdot x\right) \cdot a - z \cdot \left(b \cdot c\right)\right)\\ \end{array}\]

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

Derivation

  1. Split input into 2 regimes

  2. if a < -1.4842475539533175e+141 or 7.026026790611426e+132 < a

    1. Initial program 21.2

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

    2. Taylor expanded around -inf 11.3

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

    3. Simplified11.3

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

    if -1.4842475539533175e+141 < a < 7.026026790611426e+132

    1. Initial program 10.0

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

    3. Applied add-cube-cbrt10.3

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

    4. Applied associate-*l*10.3

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

    6. Applied sub-neg10.3

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

    7. Applied distribute-rgt-in10.3

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

    8. Applied distribute-lft-in10.3

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

    9. Simplified10.4

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

    10. Taylor expanded around inf 9.8

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

    12. Applied *-commutative9.8

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

    13. Applied associate-*r*9.4

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.4842475539533175 \cdot 10^{+141}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(b \cdot i - t \cdot x\right) \cdot a - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;a \le 7.026026790611426 \cdot 10^{+132}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(i \cdot \left(\left(-y\right) \cdot j\right) + \left(c \cdot j\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(b \cdot i - t \cdot x\right) \cdot a - z \cdot \left(b \cdot c\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 
(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)))))