Average Error: 4.1 → 3.4
Time: 45.6s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{\log \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)}}{\sin ky} \le -6.9465654213039 \cdot 10^{-310}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)\\ \mathbf{elif}\;\frac{e^{\log \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)}}{\sin ky} \le 6.94656542130387 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left(\left({kx}^{2} \cdot ky\right) \cdot \frac{1}{12} + ky\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)\\ \end{array}\]

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 (/ (exp (log (sqrt (+ (* (sin kx) (sin kx)) (* (sin ky) (sin ky)))))) (sin ky)) < -6.9465654213039e-310 or 6.94656542130387e-310 < (/ (exp (log (sqrt (+ (* (sin kx) (sin kx)) (* (sin ky) (sin ky)))))) (sin ky))

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    if -6.9465654213039e-310 < (/ (exp (log (sqrt (+ (* (sin kx) (sin kx)) (* (sin ky) (sin ky)))))) (sin ky)) < 6.94656542130387e-310

    1. Initial program 62.4

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

    2. Initial simplification62.4

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

    3. Taylor expanded around 0 49.6

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\log \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)}}{\sin ky} \le -6.9465654213039 \cdot 10^{-310}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)\\ \mathbf{elif}\;\frac{e^{\log \left(\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right)}}{\sin ky} \le 6.94656542130387 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left(\left({kx}^{2} \cdot ky\right) \cdot \frac{1}{12} + ky\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 45.6s)Debug logProfile

herbie shell --seed 2018216 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))