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

Error

Bits error versus p

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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
  3. Applied add-cbrt-cube12.4

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{p \cdot x}{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)\right) \cdot \left(\left(x - y\right) \cdot \left(x - y\right)\right)}}}}}\right)}\]
  4. Applied add-cbrt-cube24.2

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