Average Error: 3.8 → 3.2
Time: 33.9s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9999999993477511:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \left(\sqrt[3]{{\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin ky\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin ky\right)}^{2}}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 - {kx}^{2} \cdot \frac{1}{6}\right)\\ \end{array}\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Try it out

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)))) < 0.9999999993477511

    1. Initial program 2.5

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

    3. Applied add-cube-cbrt2.8

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

    if 0.9999999993477511 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))

    1. Initial program 9.0

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

    3. Applied add-sqr-sqrt9.0

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

    4. Applied sqrt-prod9.5

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

    5. Applied *-un-lft-identity9.5

      \[\leadsto \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\]

    6. Applied times-frac9.5

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \frac{\sin ky}{\sqrt{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\]

    7. Taylor expanded around 0 4.6

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9999999993477511:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \left(\sqrt[3]{{\left(\sin ky\right)}^{2}} \cdot \sqrt[3]{{\left(\sin ky\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin ky\right)}^{2}}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 - {kx}^{2} \cdot \frac{1}{6}\right)\\ \end{array}\]

Runtime

Time bar (total: 33.9s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))