Average Error: 3.7 → 3.5
Time: 36.8s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{e^{\log \left(\sqrt{{\left(\sqrt[3]{\sin kx}\right)}^{2} \cdot \left(\sin kx \cdot \sqrt[3]{\sin kx}\right) + {\left(\sin ky\right)}^{2}}\right)}} \le 1.0000000000000007:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \sqrt[3]{{\left(\sin ky\right)}^{2}} \cdot \left(\sqrt[3]{{\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin ky\right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left(\frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}}\\ \end{array}\]

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) (exp (log (sqrt (+ (* (* (cbrt (sin kx)) (sin kx)) (pow (cbrt (sin kx)) 2)) (pow (sin ky) 2)))))) < 1.0000000000000007

    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-cbrt2.4

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

    if 1.0000000000000007 < (/ (sin ky) (exp (log (sqrt (+ (* (* (cbrt (sin kx)) (sin kx)) (pow (cbrt (sin kx)) 2)) (pow (sin ky) 2))))))

    1. Initial program 36.3

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

    2. Initial simplification38.6

      \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}}\]

    3. Taylor expanded around 0 24.8

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{e^{\log \left(\sqrt{{\left(\sqrt[3]{\sin kx}\right)}^{2} \cdot \left(\sin kx \cdot \sqrt[3]{\sin kx}\right) + {\left(\sin ky\right)}^{2}}\right)}} \le 1.0000000000000007:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \sqrt[3]{{\left(\sin ky\right)}^{2}} \cdot \left(\sqrt[3]{{\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin ky\right)}^{2}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left(\frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}}\\ \end{array}\]

Runtime

Time bar (total: 36.8s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))