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

# Try it out

Results

 In Out
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

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

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