Average Error: 11.7 → 10.5
Time: 1.1m
Precision: 64
Internal Precision: 384
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.28083712820814 \cdot 10^{-23}:\\ \;\;\;\;(y \cdot \left(x \cdot z - j \cdot i\right) + \left(\left(j \cdot t\right) \cdot c\right))_* + (b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(\left(-t\right) \cdot x\right) \cdot a\right))_*\\ \mathbf{if}\;y \le -1.0475509300723086 \cdot 10^{-296}:\\ \;\;\;\;(\left(c \cdot t - y \cdot i\right) \cdot j + \left((a \cdot \left(b \cdot i - x \cdot t\right) + \left(\left(b \cdot c\right) \cdot \left(-z\right)\right))_*\right))_*\\ \mathbf{if}\;y \le 2.196590996784399 \cdot 10^{-270}:\\ \;\;\;\;(y \cdot \left(x \cdot z - j \cdot i\right) + \left(\left(j \cdot t\right) \cdot c\right))_* + (b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(\left(-t\right) \cdot x\right) \cdot a\right))_*\\ \mathbf{if}\;y \le 969.7353266691903:\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot z + (b \cdot \left(a \cdot i - z \cdot c\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(x \cdot z - j \cdot i\right) + \left(\left(j \cdot t\right) \cdot c\right))_* + \left(\sqrt[3]{(b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*} \cdot \sqrt[3]{(b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*}\right) \cdot \sqrt[3]{(b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*}\\ \end{array}\]

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 4 regimes

  2. if y < -8.28083712820814e-23 or -1.0475509300723086e-296 < y < 2.196590996784399e-270

    1. Initial program 14.7

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

    3. Applied sub-neg14.7

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

    4. Applied distribute-lft-in14.7

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

    5. Applied associate--l+14.7

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

    6. Applied simplify14.8

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

    7. Taylor expanded around inf 12.5

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

    8. Applied simplify9.5

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

    10. Applied associate-*r*8.8

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

    if -8.28083712820814e-23 < y < -1.0475509300723086e-296

    1. Initial program 8.8

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

    2. Taylor expanded around inf 15.0

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

    3. Applied simplify14.5

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

    if 2.196590996784399e-270 < y < 969.7353266691903

    1. Initial program 8.4

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

    3. Applied sub-neg8.4

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

    4. Applied distribute-lft-in8.4

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

    5. Applied associate--l+8.4

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

    6. Applied simplify9.8

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

    8. Applied associate-*r*9.6

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

    if 969.7353266691903 < y

    1. Initial program 16.0

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

    3. Applied sub-neg16.0

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

    4. Applied distribute-lft-in16.0

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

    5. Applied associate--l+16.0

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

    6. Applied simplify16.4

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

    7. Taylor expanded around inf 13.1

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

    8. Applied simplify7.8

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

    10. Applied add-cube-cbrt8.1

      \[\leadsto (y \cdot \left(x \cdot z - j \cdot i\right) + \left(\left(j \cdot t\right) \cdot c\right))_* + \color{blue}{\left(\sqrt[3]{(b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*} \cdot \sqrt[3]{(b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*}\right) \cdot \sqrt[3]{(b \cdot \left(i \cdot a - c \cdot z\right) + \left(\left(-t\right) \cdot \left(x \cdot a\right)\right))_*}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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)))))