Average Error: 7.7 → 0.0
Time: 2.2m
Precision: 64
Internal Precision: 320
$\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{p \cdot x}{\left(x - y\right) \cdot \left(x - y\right)}}}\right)}$
$\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \sqrt[3]{{\left(\frac{p}{x - y} \cdot \frac{x}{x - y}\right)}^{3}}}}\right)}$

# Try it out

Results

 In Out
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# Derivation

1. Initial program 7.7

$\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{p \cdot x}{\left(x - y\right) \cdot \left(x - y\right)}}}\right)}$
2. Using strategy rm

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{p \cdot x}{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)}}}}}\right)}$

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\color{blue}{\sqrt[3]{\left(\left(p \cdot x\right) \cdot \left(p \cdot x\right)\right) \cdot \left(p \cdot x\right)}}}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)}}}}\right)}$
5. Applied cbrt-undiv24.2

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\sqrt[3]{\frac{\left(\left(p \cdot x\right) \cdot \left(p \cdot x\right)\right) \cdot \left(p \cdot x\right)}{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)}}}}}\right)}$
6. Applied simplify0.0

$\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \sqrt[3]{\color{blue}{{\left(\frac{p}{x - y} \cdot \frac{x}{x - y}\right)}^{3}}}}}\right)}$

# Runtime

Time bar (total: 2.2m)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (p x y)
:name "sqrt(.5*(1 + 1/sqrt(1+p*x/((x-y)*(x-y)))))"
(sqrt (* 0.5 (+ 1 (/ 1 (sqrt (+ 1 (/ (* p x) (* (- x y) (- x y))))))))))