Average Error: 24.9 → 12.6
Time: 32.4s
Precision: 64
Internal Precision: 1344
${\left(x + y\right)}^{4} - {\left(\left(x + y\right) - 1\right)}^{4}$
$\begin{array}{l} \mathbf{if}\;x \le -61204.86417145002 \lor \neg \left(x \le 25327.048769209945\right):\\ \;\;\;\;\left(4 \cdot x - 6\right) \cdot \left(x \cdot x\right) + 4 \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(x + y\right)}^{4} - \left(\sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}} \cdot \sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}}\right) \cdot \sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}}\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < -61204.86417145002 or 25327.048769209945 < x

1. Initial program 59.3

${\left(x + y\right)}^{4} - {\left(\left(x + y\right) - 1\right)}^{4}$
2. Taylor expanded around inf 8.7

$\leadsto \color{blue}{\left(4 \cdot {x}^{3} + 4 \cdot x\right) - 6 \cdot {x}^{2}}$
3. Simplified8.9

$\leadsto \color{blue}{4 \cdot x + \left(4 \cdot x - 6\right) \cdot \left(x \cdot x\right)}$

## if -61204.86417145002 < x < 25327.048769209945

1. Initial program 13.8

${\left(x + y\right)}^{4} - {\left(\left(x + y\right) - 1\right)}^{4}$
2. Using strategy rm

$\leadsto {\left(x + y\right)}^{4} - \color{blue}{\left(\sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}} \cdot \sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}}\right) \cdot \sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}}}$
3. Recombined 2 regimes into one program.
4. Final simplification12.6

$\leadsto \begin{array}{l} \mathbf{if}\;x \le -61204.86417145002 \lor \neg \left(x \le 25327.048769209945\right):\\ \;\;\;\;\left(4 \cdot x - 6\right) \cdot \left(x \cdot x\right) + 4 \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(x + y\right)}^{4} - \left(\sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}} \cdot \sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}}\right) \cdot \sqrt[3]{{\left(\left(x + y\right) - 1\right)}^{4}}\\ \end{array}$

# Runtime

Time bar (total: 32.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x y)
:name "(x+y)^4-(x+y-1)^4"
(- (pow (+ x y) 4) (pow (- (+ x y) 1) 4)))