Average Error: 11.7 → 10.1
Time: 2.1m
Precision: 64
Internal Precision: 384
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.529411030259974 \cdot 10^{-09}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{if}\;b \le 5.077720195349644 \cdot 10^{+55}:\\ \;\;\;\;\left(i \cdot \left(a \cdot b\right) + \left(t \cdot x\right) \cdot \left(-a\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) - z \cdot \left(c \cdot b - y \cdot x\right)\right)\\ \mathbf{if}\;b \le +\infty:\\ \;\;\;\;\left(\left(-t\right) \cdot \left(x \cdot a\right) + \left(y \cdot x\right) \cdot z\right) - b \cdot \left(z \cdot c - i \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-t\right) \cdot \left(x \cdot a\right) + \left(y \cdot x\right) \cdot z\right) - b \cdot \left(z \cdot c - i \cdot a\right)\\ \end{array}\]

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 < -5.529411030259974e-09

    1. Initial program 8.0

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

    3. Applied add-cube-cbrt8.3

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(\left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)}\]

    4. Applied associate-*r*8.3

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}}\]

    if -5.529411030259974e-09 < b < 5.077720195349644e+55

    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

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

    4. Applied distribute-lft-in13.7

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

    5. Taylor expanded around inf 13.5

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

    6. Applied simplify14.3

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

    7. Taylor expanded around inf 12.4

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

    8. Applied simplify8.8

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

    if 5.077720195349644e+55 < b

    1. Initial program 7.5

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

    3. Applied sub-neg7.5

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

    4. Applied distribute-lft-in7.5

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

    6. Applied distribute-lft-neg-in7.5

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

    7. Applied associate-*r*8.7

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

    8. Taylor expanded around 0 18.9

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

    9. Applied simplify17.9

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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)))))