Average Error: 11.7 → 10.1
Time: 48.3s
Precision: 64
Internal Precision: 320
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.9749168121121723 \cdot 10^{-48} \lor \neg \left(y \le 7.827987889564857 \cdot 10^{+37}\right):\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(t \cdot \left(c \cdot j\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(c \cdot t\right) \cdot j\right)\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if y < -1.9749168121121723e-48 or 7.827987889564857e+37 < y

    1. Initial program 15.8

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

    3. Applied sub-neg15.8

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

    4. Applied distribute-lft-in15.8

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

    5. Taylor expanded around inf 15.6

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

    7. Applied distribute-lft-neg-in15.6

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

    8. Applied associate-*r*12.0

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

    if -1.9749168121121723e-48 < y < 7.827987889564857e+37

    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. Using strategy rm

    3. Applied sub-neg8.8

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

    4. Applied distribute-lft-in8.8

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

    5. Taylor expanded around -inf 8.8

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

    6. Simplified8.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \color{blue}{\left(y \cdot j\right) \cdot \left(-i\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.9749168121121723 \cdot 10^{-48} \lor \neg \left(y \le 7.827987889564857 \cdot 10^{+37}\right):\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(t \cdot \left(c \cdot j\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(c \cdot t\right) \cdot j\right)\\ \end{array}\]

Runtime

Time bar (total: 48.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes11.810.16.05.829.3%
herbie shell --seed 2018263 
(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)))))