Average Error: 21.1 → 9.9
Time: 2.9m
Precision: 64
Internal precision: 128
\[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
⬇
\[\begin{array}{l} \mathbf{if}\;y \le -97906.53507359796:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot {\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3.0}\right)}\right)}^3 + \sin y \cdot \sin \left({\left(\sqrt[3]{\frac{z \cdot t}{3.0}}\right)}^3\right)\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{if}\;y \le 361619410.9503674:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^2\right) - \frac{a}{b \cdot 3.0}\\ \mathbf{else}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left({\left({\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)}^3\right)}^3\right)\right) - \frac{a}{b \cdot 3.0}\\ \end{array}\]

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

Target

Original21.1
Comparison16.8
Herbie9.9
\[ \begin{array}{l} \mathbf{if}\;z \lt -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{if}\;z \lt 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2.0\right) \cdot \cos \left(y - \frac{t}{3.0} \cdot z\right) - \frac{\frac{a}{3.0}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2.0 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3.0}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if y < -97906.53507359796

    1. Initial program 20.8

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
    2. Using strategy rm

    3. Applied cos-diff 19.8

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right)} - \frac{a}{b \cdot 3.0}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt 19.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\frac{z \cdot t}{3.0}}\right)}^3\right)}\right) - \frac{a}{b \cdot 3.0}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt 19.9

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{{\left(\sqrt[3]{\cos \left(\frac{z \cdot t}{3.0}\right)}\right)}^3} + \sin y \cdot \sin \left({\left(\sqrt[3]{\frac{z \cdot t}{3.0}}\right)}^3\right)\right) - \frac{a}{b \cdot 3.0}\]

    if -97906.53507359796 < y < 361619410.9503674

    1. Initial program 21.4

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]

    2. Applied taylor 0.1

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^2\right) - \frac{a}{b \cdot 3.0}\]

    3. Taylor expanded around 0 0.1

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^2\right)} - \frac{a}{b \cdot 3.0}\]

    if 361619410.9503674 < y

    1. Initial program 21.0

      \[\left(2.0 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3.0}\right) - \frac{a}{b \cdot 3.0}\]
    2. Using strategy rm

    3. Applied cos-diff 20.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3.0}\right)\right)} - \frac{a}{b \cdot 3.0}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt 20.1

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\frac{z \cdot t}{3.0}}\right)}^3\right)}\right) - \frac{a}{b \cdot 3.0}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt 20.0

      \[\leadsto \left(2.0 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3.0}\right) + \sin y \cdot \sin \left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{z \cdot t}{3.0}}}\right)}^3\right)}}^3\right)\right) - \frac{a}{b \cdot 3.0}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 2.9m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(31753201 2071411468 4126184347 151412706 3360758988 386482930)'
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))