Average Error: 3.8 → 3.2
Time: 20.8s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1.0:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin ky}\right)}^{2} \cdot \left(\sin ky \cdot \sqrt[3]{\sin ky}\right) + {\left(\sin kx\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(\left({kx}^{2} \cdot ky\right) \cdot \frac{1}{12} + ky\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \end{array}\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) < 1.0

    1. Initial program 2.0

      \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
    2. Using strategy rm

    3. Applied add-cube-cbrt2.6

      \[\leadsto \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\color{blue}{\left(\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right) \cdot \sqrt[3]{\sin ky}\right)}}^{2}}} \cdot \sin th\]

    4. Applied unpow-prod-down2.6

      \[\leadsto \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \color{blue}{{\left(\sqrt[3]{\sin ky} \cdot \sqrt[3]{\sin ky}\right)}^{2} \cdot {\left(\sqrt[3]{\sin ky}\right)}^{2}}}} \cdot \sin th\]

    5. Simplified2.4

      \[\leadsto \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \color{blue}{\left(\sqrt[3]{\sin ky} \cdot \sin ky\right)} \cdot {\left(\sqrt[3]{\sin ky}\right)}^{2}}} \cdot \sin th\]

    if 1.0 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))

    1. Initial program 61.9

      \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

    2. Taylor expanded around 0 29.4

      \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}}} \cdot \sin th\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1.0:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin ky}\right)}^{2} \cdot \left(\sin ky \cdot \sqrt[3]{\sin ky}\right) + {\left(\sin kx\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(\left({kx}^{2} \cdot ky\right) \cdot \frac{1}{12} + ky\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \end{array}\]

Runtime

Time bar (total: 20.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes4.23.22.51.758.5%
herbie shell --seed 2018286 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))