Average Error: 5.4 → 3.4
Time: 26.3s
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.209969181527008 \cdot 10^{+43}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - 4.0 \cdot \left(a \cdot t\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.309960619258739 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot t\right) \cdot \left(18.0 \cdot \left(y \cdot x\right)\right) - \left(4.0 \cdot a\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(18.0 \cdot \left(z \cdot \left(y \cdot x\right)\right)\right) - 4.0 \cdot \left(a \cdot t\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \end{array}\]

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

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -6.209969181527008e+43

    1. Initial program 1.6

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Taylor expanded around 0 1.6

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(18.0 \cdot \left(x \cdot y\right)\right)} \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    3. Using strategy rm

    4. Applied associate-*l*1.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(18.0 \cdot \left(\left(x \cdot y\right) \cdot z\right)\right)} \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    5. Taylor expanded around inf 1.5

      \[\leadsto \left(\left(\left(\left(18.0 \cdot \left(\left(x \cdot y\right) \cdot z\right)\right) \cdot t - \color{blue}{4.0 \cdot \left(t \cdot a\right)}\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    6. Taylor expanded around -inf 1.2

      \[\leadsto \left(\left(\left(\color{blue}{18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} - 4.0 \cdot \left(t \cdot a\right)\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if -6.209969181527008e+43 < t < 2.309960619258739e-98

    1. Initial program 7.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Taylor expanded around 0 7.5

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(18.0 \cdot \left(x \cdot y\right)\right)} \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    3. Using strategy rm

    4. Applied associate-*l*4.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(18.0 \cdot \left(x \cdot y\right)\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if 2.309960619258739e-98 < t

    1. Initial program 3.1

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Taylor expanded around 0 3.1

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(18.0 \cdot \left(x \cdot y\right)\right)} \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
    3. Using strategy rm

    4. Applied associate-*l*3.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(18.0 \cdot \left(\left(x \cdot y\right) \cdot z\right)\right)} \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    5. Taylor expanded around inf 3.0

      \[\leadsto \left(\left(\left(\left(18.0 \cdot \left(\left(x \cdot y\right) \cdot z\right)\right) \cdot t - \color{blue}{4.0 \cdot \left(t \cdot a\right)}\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    6. Taylor expanded around 0 2.9

      \[\leadsto \left(\left(\left(\left(18.0 \cdot \left(\left(x \cdot y\right) \cdot z\right)\right) \cdot t - 4.0 \cdot \left(t \cdot a\right)\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \color{blue}{27.0 \cdot \left(j \cdot k\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.209969181527008 \cdot 10^{+43}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - 4.0 \cdot \left(a \cdot t\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.309960619258739 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot t\right) \cdot \left(18.0 \cdot \left(y \cdot x\right)\right) - \left(4.0 \cdot a\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(18.0 \cdot \left(z \cdot \left(y \cdot x\right)\right)\right) - 4.0 \cdot \left(a \cdot t\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot k\right) \cdot 27.0\\ \end{array}\]

Runtime

Time bar (total: 26.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes5.33.40.74.541.7%
herbie shell --seed 2018340 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))