Average Error: 5.4 → 1.7
Time: 1.3m
Precision: 64
Internal Precision: 384
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.3924307365786572 \cdot 10^{-27}:\\ \;\;\;\;(\left(z \cdot 18.0\right) \cdot \left(\left(t \cdot y\right) \cdot x\right) + \left(c \cdot b\right))_* - (4.0 \cdot \left((a \cdot t + \left(x \cdot i\right))_*\right) + \left(\left(k \cdot j\right) \cdot 27.0\right))_*\\ \mathbf{if}\;z \le 5.439204050419734 \cdot 10^{-133}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot \left(y \cdot z\right)\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(\left(z \cdot 18.0\right) \cdot \left(t \cdot \left(y \cdot x\right)\right) + \left(c \cdot b\right))_* - (4.0 \cdot \left((a \cdot t + \left(x \cdot i\right))_*\right) + \left(\left(k \cdot j\right) \cdot 27.0\right))_*\\ \end{array}\]

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

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if z < -1.3924307365786572e-27

    1. Initial program 6.8

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

    2. Taylor expanded around inf 2.1

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

    3. Applied simplify1.9

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

    5. Applied associate-*r*1.8

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

    if -1.3924307365786572e-27 < z < 5.439204050419734e-133

    1. Initial program 4.4

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

    3. Applied associate-*l*0.9

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

    if 5.439204050419734e-133 < z

    1. Initial program 5.8

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

    2. Taylor expanded around inf 3.0

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

    3. Applied simplify2.8

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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)))