Average Error: 11.6 → 9.4
Time: 2.2m
Precision: 64
Internal Precision: 384
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -1.8701855460597052 \cdot 10^{-165}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j - b \cdot z\right) + \left(z \cdot y - a \cdot t\right) \cdot x\right) - \left(i \cdot \left(j \cdot y\right) + \left(-b\right) \cdot \left(i \cdot a\right)\right)\\ \mathbf{if}\;c \le 9.899517850141313 \cdot 10^{-257}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z\right) + \left(b \cdot \left(-i\right)\right) \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{if}\;c \le 8.786293772213101 \cdot 10^{-07}:\\ \;\;\;\;\left(\left(z \cdot y - a \cdot t\right) \cdot x - \left(z \cdot c\right) \cdot b\right) - \left(\left(b \cdot a\right) \cdot \left(-i\right) - j \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j - b \cdot z\right) + \left(z \cdot y - a \cdot t\right) \cdot x\right) - \left(i \cdot \left(j \cdot y\right) + \left(-b\right) \cdot \left(i \cdot a\right)\right)\\ \end{array}\]

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 c < -1.8701855460597052e-165 or 8.786293772213101e-07 < c

    1. Initial program 13.7

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

    3. Applied sub-neg13.7

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

    4. Applied distribute-lft-in13.7

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

    5. Taylor expanded around inf 13.5

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

    6. Applied simplify9.5

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

    if -1.8701855460597052e-165 < c < 9.899517850141313e-257

    1. Initial program 8.7

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

    3. Applied sub-neg8.7

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

    4. Applied distribute-lft-in8.7

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

    6. Applied distribute-lft-neg-in8.7

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

    7. Applied associate-*r*9.4

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

    if 9.899517850141313e-257 < c < 8.786293772213101e-07

    1. Initial program 9.0

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

    3. Applied sub-neg9.0

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

    4. Applied distribute-lft-in9.0

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

    5. Taylor expanded around inf 9.0

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

    6. Applied simplify9.1

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(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)))))