Average Error: 23.3 → 0.1
Time: 30.3s
Precision: 64
Internal Precision: 576
\[\frac{x \cdot x + 1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000}\]
\[\begin{array}{l} \mathbf{if}\;\left(\frac{1}{{x}^{4}} - \frac{100000}{{x}^{6}}\right) + \frac{\frac{1}{x}}{x} \le -3.9301041951962 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1 + x \cdot x}{100000 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)}^{3}}\\ \mathbf{if}\;\left(\frac{1}{{x}^{4}} - \frac{100000}{{x}^{6}}\right) + \frac{\frac{1}{x}}{x} \le 3.2483317770887697 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{1}{{x}^{4}} - \frac{100000}{{x}^{6}}\right) + \frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x + 1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if (+ (- (/ 1 (pow x 4)) (/ 100000 (pow x 6))) (/ (/ 1 x) x)) < -3.9301041951962e-310

    1. Initial program 0.0

      \[\frac{x \cdot x + 1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000}\]
    2. Using strategy rm
    3. Applied add-cbrt-cube0.1

      \[\leadsto \frac{x \cdot x + 1}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right)}}}\]
    4. Applied add-cbrt-cube0.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + 1\right) \cdot \left(x \cdot x + 1\right)\right) \cdot \left(x \cdot x + 1\right)}}}{\sqrt[3]{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right)}}\]
    5. Applied cbrt-undiv1.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x \cdot x + 1\right) \cdot \left(x \cdot x + 1\right)\right) \cdot \left(x \cdot x + 1\right)}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000\right)}}}\]
    6. Applied simplify0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1 + x \cdot x}{100000 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)}^{3}}}\]

    if -3.9301041951962e-310 < (+ (- (/ 1 (pow x 4)) (/ 100000 (pow x 6))) (/ (/ 1 x) x)) < 3.2483317770887697e-21

    1. Initial program 47.9

      \[\frac{x \cdot x + 1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000}\]
    2. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{2}} + \frac{1}{{x}^{4}}\right) - 100000 \cdot \frac{1}{{x}^{6}}}\]
    3. Applied simplify0.1

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{4}} - \frac{100000}{{x}^{6}}\right) + \frac{\frac{1}{x}}{x}}\]

    if 3.2483317770887697e-21 < (+ (- (/ 1 (pow x 4)) (/ 100000 (pow x 6))) (/ (/ 1 x) x))

    1. Initial program 0.0

      \[\frac{x \cdot x + 1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x + 100000}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 30.3s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "(x*x+1)/(x*x*x*x+100000)"
  (/ (+ (* x x) 1) (+ (* (* (* x x) x) x) 100000)))