Average Error: 11.5 → 11.5
Time: 43.4s
Precision: 64
Internal Precision: 576
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.401840283372464 \cdot 10^{-179} \lor \neg \left(x \le 4.8917016549891425 \cdot 10^{-222}\right):\\ \;\;\;\;\left((y \cdot z + \left(a \cdot \left(-t\right)\right))_* \cdot x - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z - i \cdot a\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

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 2 regimes

  2. if x < -5.401840283372464e-179 or 4.8917016549891425e-222 < x

    1. Initial program 9.9

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

    3. Applied prod-diff9.9

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

    4. Applied distribute-lft-in9.9

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

    5. Simplified9.9

      \[\leadsto \left(\left(x \cdot (y \cdot z + \left(-a \cdot t\right))_* + \color{blue}{0}\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

    if -5.401840283372464e-179 < x < 4.8917016549891425e-222

    1. Initial program 17.3

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

    2. Taylor expanded around 0 17.2

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.401840283372464 \cdot 10^{-179} \lor \neg \left(x \le 4.8917016549891425 \cdot 10^{-222}\right):\\ \;\;\;\;\left((y \cdot z + \left(a \cdot \left(-t\right)\right))_* \cdot x - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z - i \cdot a\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \end{array}\]

Runtime

Time bar (total: 43.4s)Debug logProfile

herbie shell --seed 2018220 +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)))))