Average Error: 11.3 → 11.2
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.034411174810758 \cdot 10^{-118}:\\ \;\;\;\;\left(z \cdot y - a \cdot t\right) \cdot x - \left(\left(z \cdot c - i \cdot a\right) \cdot b - \left(\left(t \cdot j\right) \cdot c + \left(j \cdot i\right) \cdot \left(-y\right)\right)\right)\\ \mathbf{if}\;b \le -2.520365337709809 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(t \cdot c\right) \cdot j + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \left(j \cdot y - a \cdot b\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(z \cdot x\right) + \left(i \cdot y\right) \cdot \left(-j\right)\right) - \left(t \cdot \left(x \cdot a - j \cdot c\right) + b \cdot \left(z \cdot c - i \cdot a\right)\right)\\ \end{array}\]

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 b < -6.034411174810758e-118

    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

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

    4. Applied distribute-lft-in8.4

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

    6. Applied associate-*r*8.6

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

    7. Taylor expanded around inf 8.6

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

    8. Applied simplify8.7

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

    if -6.034411174810758e-118 < b < -2.520365337709809e-293

    1. Initial program 15.9

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

    3. Applied sub-neg15.9

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

    4. Applied distribute-lft-in15.9

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

    6. Applied associate-*r*16.3

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

    7. Taylor expanded around inf 16.2

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

    8. Applied simplify15.7

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

    9. Taylor expanded around inf 17.9

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

    10. Applied simplify13.5

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

    if -2.520365337709809e-293 < b

    1. Initial program 11.4

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

    3. Applied sub-neg11.4

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

    4. Applied distribute-lft-in11.4

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

    5. Taylor expanded around inf 11.6

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

    6. Applied simplify11.9

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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)))))