Average Error: 11.3 → 11.4
Time: 58.0s
Precision: 64
Internal Precision: 320
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -5.388035524830058 \cdot 10^{-175} \lor \neg \left(j \le 1.8786470981504728 \cdot 10^{-206}\right):\\ \;\;\;\;(\left((\left(-a\right) \cdot t + \left(y \cdot z\right))_*\right) \cdot x + \left((\left((\left(-i\right) \cdot a + \left(c \cdot z\right))_*\right) \cdot \left(-b\right) + \left(j \cdot \left(t \cdot c - i \cdot y\right)\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left((\left(-a\right) \cdot t + \left(y \cdot z\right))_*\right) \cdot x + \left((\left((\left(-i\right) \cdot a + \left(c \cdot z\right))_*\right) \cdot \left(-b\right) + 0)_*\right))_*\\ \end{array}\]

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 j < -5.388035524830058e-175 or 1.8786470981504728e-206 < j

    1. Initial program 9.7

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

    2. Initial simplification9.7

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

    if -5.388035524830058e-175 < j < 1.8786470981504728e-206

    1. Initial program 16.7

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

    2. Initial simplification16.7

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

    3. Taylor expanded around 0 17.1

      \[\leadsto (\left((\left(-a\right) \cdot t + \left(z \cdot y\right))_*\right) \cdot x + \left((\left((\left(-i\right) \cdot a + \left(z \cdot c\right))_*\right) \cdot \left(-b\right) + \color{blue}{0})_*\right))_*\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -5.388035524830058 \cdot 10^{-175} \lor \neg \left(j \le 1.8786470981504728 \cdot 10^{-206}\right):\\ \;\;\;\;(\left((\left(-a\right) \cdot t + \left(y \cdot z\right))_*\right) \cdot x + \left((\left((\left(-i\right) \cdot a + \left(c \cdot z\right))_*\right) \cdot \left(-b\right) + \left(j \cdot \left(t \cdot c - i \cdot y\right)\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left((\left(-a\right) \cdot t + \left(y \cdot z\right))_*\right) \cdot x + \left((\left((\left(-i\right) \cdot a + \left(c \cdot z\right))_*\right) \cdot \left(-b\right) + 0)_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 58.0s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(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)))))