Average Error: 23.5 → 18.8
Time: 15.6s
Precision: 64
Internal Precision: 2624
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
\[\begin{array}{l} \mathbf{if}\;x \le 6.294595312968477 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{0.5 + \log \left(e^{0.5 \cdot \frac{x}{\sqrt{x \cdot x + 4 \cdot \left(p \cdot p\right)}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5 \cdot x}{x} + 0.5}\\ \end{array}\]

Error

Bits error versus x

Bits error versus p

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < 6.294595312968477e+146

    1. Initial program 20.5

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
    2. Initial simplification20.6

      \[\leadsto \sqrt{\frac{0.5 \cdot x}{\sqrt{\left(p \cdot p\right) \cdot 4 + x \cdot x}} + 0.5}\]
    3. Using strategy rm
    4. Applied *-un-lft-identity20.6

      \[\leadsto \sqrt{\frac{0.5 \cdot x}{\color{blue}{1 \cdot \sqrt{\left(p \cdot p\right) \cdot 4 + x \cdot x}}} + 0.5}\]
    5. Applied times-frac20.6

      \[\leadsto \sqrt{\color{blue}{\frac{0.5}{1} \cdot \frac{x}{\sqrt{\left(p \cdot p\right) \cdot 4 + x \cdot x}}} + 0.5}\]
    6. Simplified20.6

      \[\leadsto \sqrt{\color{blue}{0.5} \cdot \frac{x}{\sqrt{\left(p \cdot p\right) \cdot 4 + x \cdot x}} + 0.5}\]
    7. Using strategy rm
    8. Applied add-log-exp20.6

      \[\leadsto \sqrt{\color{blue}{\log \left(e^{0.5 \cdot \frac{x}{\sqrt{\left(p \cdot p\right) \cdot 4 + x \cdot x}}}\right)} + 0.5}\]

    if 6.294595312968477e+146 < x

    1. Initial program 43.0

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
    2. Initial simplification43.0

      \[\leadsto \sqrt{\frac{0.5 \cdot x}{\sqrt{\left(p \cdot p\right) \cdot 4 + x \cdot x}} + 0.5}\]
    3. Taylor expanded around 0 7.3

      \[\leadsto \sqrt{\frac{0.5 \cdot x}{\color{blue}{x}} + 0.5}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 6.294595312968477 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{0.5 + \log \left(e^{0.5 \cdot \frac{x}{\sqrt{x \cdot x + 4 \cdot \left(p \cdot p\right)}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5 \cdot x}{x} + 0.5}\\ \end{array}\]

Runtime

Time bar (total: 15.6s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x p)
  :name "sqrt( 0.5 * (1 + x/(sqrt(4*p*p +x*x))))"
  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))