Average Error: 35.8 → 0.4
Time: 17.6s
Precision: 64
Internal Precision: 2368
\[\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}\]
\[\begin{array}{l} \mathbf{if}\;a \le -0.043473338976223706:\\ \;\;\;\;\frac{1}{a \cdot a} \cdot \frac{a - \sin a}{a}\\ \mathbf{elif}\;a \le 0.046591036094757086:\\ \;\;\;\;\left(\frac{1}{5040} \cdot {a}^{4} + \frac{1}{6}\right) - \frac{1}{120} \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}} \cdot \sqrt[3]{\sqrt[3]{a - \sin a}}}{a}\\ \end{array}\]

Error

Bits error versus a

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if a < -0.043473338976223706

    1. Initial program 10.3

      \[\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}\]
    2. Initial simplification10.3

      \[\leadsto \frac{a - \sin a}{{a}^{3}}\]
    3. Using strategy rm
    4. Applied unpow310.3

      \[\leadsto \frac{a - \sin a}{\color{blue}{\left(a \cdot a\right) \cdot a}}\]
    5. Applied *-un-lft-identity10.3

      \[\leadsto \frac{\color{blue}{1 \cdot \left(a - \sin a\right)}}{\left(a \cdot a\right) \cdot a}\]
    6. Applied times-frac0.7

      \[\leadsto \color{blue}{\frac{1}{a \cdot a} \cdot \frac{a - \sin a}{a}}\]

    if -0.043473338976223706 < a < 0.046591036094757086

    1. Initial program 61.2

      \[\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}\]
    2. Initial simplification61.2

      \[\leadsto \frac{a - \sin a}{{a}^{3}}\]
    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{6} + \frac{1}{5040} \cdot {a}^{4}\right) - \frac{1}{120} \cdot {a}^{2}}\]

    if 0.046591036094757086 < a

    1. Initial program 11.0

      \[\frac{a - \sin a}{\left(a \cdot a\right) \cdot a}\]
    2. Initial simplification11.0

      \[\leadsto \frac{a - \sin a}{{a}^{3}}\]
    3. Using strategy rm
    4. Applied add-cube-cbrt11.3

      \[\leadsto \frac{a - \sin a}{\color{blue}{\left(\sqrt[3]{{a}^{3}} \cdot \sqrt[3]{{a}^{3}}\right) \cdot \sqrt[3]{{a}^{3}}}}\]
    5. Applied add-cube-cbrt11.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}\right) \cdot \sqrt[3]{a - \sin a}}}{\left(\sqrt[3]{{a}^{3}} \cdot \sqrt[3]{{a}^{3}}\right) \cdot \sqrt[3]{{a}^{3}}}\]
    6. Applied times-frac11.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}}{\sqrt[3]{{a}^{3}} \cdot \sqrt[3]{{a}^{3}}} \cdot \frac{\sqrt[3]{a - \sin a}}{\sqrt[3]{{a}^{3}}}}\]
    7. Simplified11.3

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right)} \cdot \frac{\sqrt[3]{a - \sin a}}{\sqrt[3]{{a}^{3}}}\]
    8. Simplified0.7

      \[\leadsto \left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \color{blue}{\frac{\sqrt[3]{a - \sin a}}{a}}\]
    9. Using strategy rm
    10. Applied add-cube-cbrt0.7

      \[\leadsto \left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}\right) \cdot \sqrt[3]{a - \sin a}}}}{a}\]
    11. Applied cbrt-prod0.7

      \[\leadsto \left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}} \cdot \sqrt[3]{\sqrt[3]{a - \sin a}}}}{a}\]
  3. Recombined 3 regimes into one program.
  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.043473338976223706:\\ \;\;\;\;\frac{1}{a \cdot a} \cdot \frac{a - \sin a}{a}\\ \mathbf{elif}\;a \le 0.046591036094757086:\\ \;\;\;\;\left(\frac{1}{5040} \cdot {a}^{4} + \frac{1}{6}\right) - \frac{1}{120} \cdot {a}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a - \sin a}}{a} \cdot \frac{\sqrt[3]{a - \sin a}}{a}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a - \sin a} \cdot \sqrt[3]{a - \sin a}} \cdot \sqrt[3]{\sqrt[3]{a - \sin a}}}{a}\\ \end{array}\]

Runtime

Time bar (total: 17.6s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (a)
  :name "(a - sin(a))/(a*a*a)"
  (/ (- a (sin a)) (* (* a a) a)))