Average Error: 33.7 → 29.1
Time: 33.7s
Precision: 64
Internal Precision: 3392
\[y \cdot \sin \left(\frac{\frac{pi}{4} \cdot x}{y}\right)\]
\[\begin{array}{l} \mathbf{if}\;y \cdot \log \left(e^{\sin \left(\frac{\frac{pi}{4} \cdot x}{y}\right)}\right) \le -1.1080319782796545 \cdot 10^{-286}:\\ \;\;\;\;y \cdot \sin \left(\frac{\left(\sqrt[3]{\frac{pi}{4} \cdot x} \cdot \sqrt[3]{\frac{pi}{4} \cdot x}\right) \cdot \sqrt[3]{\frac{pi}{4} \cdot x}}{y}\right)\\ \mathbf{if}\;y \cdot \log \left(e^{\sin \left(\frac{\frac{pi}{4} \cdot x}{y}\right)}\right) \le 1.449412991445007 \cdot 10^{-306}:\\ \;\;\;\;x \cdot \left(pi \cdot \frac{1}{4}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sin \left(\left(\sqrt[3]{\frac{\frac{pi}{4} \cdot x}{y}} \cdot \sqrt[3]{\frac{\frac{pi}{4} \cdot x}{y}}\right) \cdot \sqrt[3]{\frac{\frac{pi}{4} \cdot x}{y}}\right)\\ \end{array}\]

Error

Bits error versus y

Bits error versus pi

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes
  2. if (* y (log (exp (sin (/ (* (/ pi 4) x) y))))) < -1.1080319782796545e-286

    1. Initial program 51.5

      \[y \cdot \sin \left(\frac{\frac{pi}{4} \cdot x}{y}\right)\]
    2. Using strategy rm
    3. Applied add-cube-cbrt51.7

      \[\leadsto y \cdot \sin \left(\frac{\color{blue}{\left(\sqrt[3]{\frac{pi}{4} \cdot x} \cdot \sqrt[3]{\frac{pi}{4} \cdot x}\right) \cdot \sqrt[3]{\frac{pi}{4} \cdot x}}}{y}\right)\]

    if -1.1080319782796545e-286 < (* y (log (exp (sin (/ (* (/ pi 4) x) y))))) < 1.449412991445007e-306

    1. Initial program 10.4

      \[y \cdot \sin \left(\frac{\frac{pi}{4} \cdot x}{y}\right)\]
    2. Taylor expanded around inf 10.5

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{4} \cdot \frac{x \cdot pi}{y}\right)}\]
    3. Applied simplify1.0

      \[\leadsto \color{blue}{x \cdot \left(pi \cdot \frac{1}{4}\right)}\]

    if 1.449412991445007e-306 < (* y (log (exp (sin (/ (* (/ pi 4) x) y)))))

    1. Initial program 57.7

      \[y \cdot \sin \left(\frac{\frac{pi}{4} \cdot x}{y}\right)\]
    2. Using strategy rm
    3. Applied add-cube-cbrt57.7

      \[\leadsto y \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{pi}{4} \cdot x}{y}} \cdot \sqrt[3]{\frac{\frac{pi}{4} \cdot x}{y}}\right) \cdot \sqrt[3]{\frac{\frac{pi}{4} \cdot x}{y}}\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 33.7s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (y pi x)
  :name "y*sin(pi/4*x/y)"
  (* y (sin (/ (* (/ pi 4) x) y))))