Average Error: 32.9 → 19.8
Time: 1.7m
Precision: 64
Internal Precision: 320
\[\frac{x - y}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\]
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}} \le -5.68878655196326 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}} \cdot \sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}} \cdot \frac{x - y}{\sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}}\\ \mathbf{if}\;\frac{x - y}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}} \le 7.187370790730964 \cdot 10^{-272}:\\ \;\;\;\;\frac{x - y}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x - y}}{\sqrt{1}} \cdot \frac{\sqrt{x - y}}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus p

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if (/ (- x y) (sqrt (+ (* p x) (pow (- x y) 2)))) < -5.68878655196326e-310

    1. Initial program 2.8

      \[\frac{x - y}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\]
    2. Using strategy rm
    3. Applied add-cube-cbrt4.0

      \[\leadsto \frac{x - y}{\color{blue}{\left(\sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}} \cdot \sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}}}\]
    4. Applied *-un-lft-identity4.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}} \cdot \sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}}\]
    5. Applied times-frac4.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}} \cdot \sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}} \cdot \frac{x - y}{\sqrt[3]{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}}}\]

    if -5.68878655196326e-310 < (/ (- x y) (sqrt (+ (* p x) (pow (- x y) 2)))) < 7.187370790730964e-272

    1. Initial program 62.0

      \[\frac{x - y}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\]
    2. Taylor expanded around 0 33.0

      \[\leadsto \frac{x - y}{\color{blue}{y - x}}\]

    if 7.187370790730964e-272 < (/ (- x y) (sqrt (+ (* p x) (pow (- x y) 2))))

    1. Initial program 11.7

      \[\frac{x - y}{\sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\]
    2. Using strategy rm
    3. Applied *-un-lft-identity11.7

      \[\leadsto \frac{x - y}{\sqrt{\color{blue}{1 \cdot \left(p \cdot x + {\left(x - y\right)}^{2}\right)}}}\]
    4. Applied sqrt-prod11.7

      \[\leadsto \frac{x - y}{\color{blue}{\sqrt{1} \cdot \sqrt{p \cdot x + {\left(x - y\right)}^{2}}}}\]
    5. Applied add-sqr-sqrt12.1

      \[\leadsto \frac{\color{blue}{\sqrt{x - y} \cdot \sqrt{x - y}}}{\sqrt{1} \cdot \sqrt{p \cdot x + {\left(x - y\right)}^{2}}}\]
    6. Applied times-frac12.1

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

Runtime

Time bar (total: 1.7m)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x y p)
  :name "(x-y)/(sqrt(p*x + (x-y)^2))"
  (/ (- x y) (sqrt (+ (* p x) (pow (- x y) 2)))))