Average Error: 30.6 → 0.3
Time: 24.2s
Precision: 64
Internal Precision: 2368
\[\frac{\sin x}{{x}^{2}} - \frac{\cos x}{x}\]
\[\begin{array}{l} \mathbf{if}\;x \le -0.04242194793463719 \lor \neg \left(x \le 0.046591036094757086\right):\\ \;\;\;\;\frac{\sqrt[3]{\sin x}}{\sqrt[3]{{x}^{2}}} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}} - \frac{\cos x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{840} \cdot {x}^{5} + \frac{1}{3} \cdot x\right) - \frac{1}{30} \cdot {x}^{3}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < -0.04242194793463719 or 0.046591036094757086 < x

    1. Initial program 0.2

      \[\frac{\sin x}{{x}^{2}} - \frac{\cos x}{x}\]
    2. Using strategy rm
    3. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\sin x}{\color{blue}{\left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \sqrt[3]{{x}^{2}}}} - \frac{\cos x}{x}\]
    4. Applied add-cube-cbrt0.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}}}{\left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \sqrt[3]{{x}^{2}}} - \frac{\cos x}{x}\]
    5. Applied times-frac0.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}} \cdot \frac{\sqrt[3]{\sin x}}{\sqrt[3]{{x}^{2}}}} - \frac{\cos x}{x}\]

    if -0.04242194793463719 < x < 0.046591036094757086

    1. Initial program 61.4

      \[\frac{\sin x}{{x}^{2}} - \frac{\cos x}{x}\]
    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\frac{1}{840} \cdot {x}^{5} + \frac{1}{3} \cdot x\right) - \frac{1}{30} \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.04242194793463719 \lor \neg \left(x \le 0.046591036094757086\right):\\ \;\;\;\;\frac{\sqrt[3]{\sin x}}{\sqrt[3]{{x}^{2}}} \cdot \frac{\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}}{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}} - \frac{\cos x}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{840} \cdot {x}^{5} + \frac{1}{3} \cdot x\right) - \frac{1}{30} \cdot {x}^{3}\\ \end{array}\]

Runtime

Time bar (total: 24.2s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "sin(x)/x^2-cos(x)/x"
  (- (/ (sin x) (pow x 2)) (/ (cos x) x)))