Average Error: 11.7 → 10.4
Time: 48.6s
Precision: 64
Internal Precision: 320
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.918032792012503 \cdot 10^{+139}:\\ \;\;\;\;j \cdot \left(t \cdot c - i \cdot y\right) + (a \cdot \left(b \cdot i - t \cdot x\right) + \left(\left(c \cdot b\right) \cdot \left(-z\right)\right))_*\\ \mathbf{elif}\;a \le -6.409151110904502 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(\left(t \cdot a\right) \cdot \left(-x\right) + z \cdot \left(y \cdot x\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(j \cdot \left(t \cdot c\right) + \left(i \cdot y\right) \cdot \left(-j\right)\right)\\ \mathbf{elif}\;a \le 1.3193342167850466 \cdot 10^{+28}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(j \cdot \left(t \cdot c\right) + \left(-i\right) \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - i \cdot y\right) + (a \cdot \left(b \cdot i - t \cdot x\right) + \left(\left(c \cdot b\right) \cdot \left(-z\right)\right))_*\\ \end{array}\]

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 3 regimes

  2. if a < -9.918032792012503e+139 or 1.3193342167850466e+28 < a

    1. Initial program 17.9

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

    2. Taylor expanded around inf 13.0

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

    3. Simplified13.0

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

    if -9.918032792012503e+139 < a < -6.409151110904502e-142

    1. Initial program 10.2

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

    3. Applied sub-neg10.2

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

    4. Applied distribute-lft-in10.2

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

    6. Applied sub-neg10.2

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

    7. Applied distribute-lft-in10.2

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

    9. Applied associate-*r*10.0

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

    if -6.409151110904502e-142 < a < 1.3193342167850466e+28

    1. Initial program 9.3

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

    3. Applied sub-neg9.3

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

    4. Applied distribute-lft-in9.3

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

    5. Taylor expanded around -inf 9.2

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

    6. Simplified9.2

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.918032792012503 \cdot 10^{+139}:\\ \;\;\;\;j \cdot \left(t \cdot c - i \cdot y\right) + (a \cdot \left(b \cdot i - t \cdot x\right) + \left(\left(c \cdot b\right) \cdot \left(-z\right)\right))_*\\ \mathbf{elif}\;a \le -6.409151110904502 \cdot 10^{-142}:\\ \;\;\;\;\left(\left(\left(t \cdot a\right) \cdot \left(-x\right) + z \cdot \left(y \cdot x\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(j \cdot \left(t \cdot c\right) + \left(i \cdot y\right) \cdot \left(-j\right)\right)\\ \mathbf{elif}\;a \le 1.3193342167850466 \cdot 10^{+28}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(j \cdot \left(t \cdot c\right) + \left(-i\right) \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(t \cdot c - i \cdot y\right) + (a \cdot \left(b \cdot i - t \cdot x\right) + \left(\left(c \cdot b\right) \cdot \left(-z\right)\right))_*\\ \end{array}\]

Runtime

Time bar (total: 48.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes11.810.41.710.113.6%
herbie shell --seed 2018263 +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)))))