Average Error: 3.9 → 3.2
Time: 38.7s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 4.9214811509019 \cdot 10^{-312}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + \left({kx}^{2} \cdot ky\right) \cdot \frac{1}{12}\right) - \frac{1}{6} \cdot {ky}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right) \cdot \sin ky\\ \end{array}\]

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 (+ (pow (sin kx) 2) (pow (sin ky) 2)) < 4.9214811509019e-312

    1. Initial program 58.8

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

    2. Taylor expanded around 0 45.2

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

    if 4.9214811509019e-312 < (+ (pow (sin kx) 2) (pow (sin ky) 2))

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    4. Applied associate-*l*0.4

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 4.9214811509019 \cdot 10^{-312}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + \left({kx}^{2} \cdot ky\right) \cdot \frac{1}{12}\right) - \frac{1}{6} \cdot {ky}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right) \cdot \sin ky\\ \end{array}\]

Runtime

Time bar (total: 38.7s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))