Average Error: 4.1 → 3.4
Time: 42.4s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky} \le -6.9465654213039 \cdot 10^{-310}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)\\ \mathbf{elif}\;\frac{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}{\sin ky} \le 6.9465654213039 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\left(ky + \left(ky \cdot {kx}^{2}\right) \cdot \frac{1}{12}\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)\\ \end{array}\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (sqrt (+ (* (sin kx) (sin kx)) (* (sin ky) (sin ky)))) (sin ky)) < -6.9465654213039e-310 or 6.9465654213039e-310 < (/ (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 < (/ (sqrt (+ (* (sin kx) (sin kx)) (* (sin ky) (sin ky)))) (sin ky)) < 6.9465654213039e-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

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

Runtime

Time bar (total: 42.4s)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)))