Average Error: 3.7 → 3.5
Time: 26.2s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} = -\infty:\\ \;\;\;\;\sin ky \cdot \left(\frac{1}{\left(ky + \frac{1}{12} \cdot \left(ky \cdot {kx}^{2}\right)\right) - {ky}^{3} \cdot \frac{1}{6}} \cdot \sin th\right)\\ \mathbf{elif}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9730656778675094:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left(ky \cdot {kx}^{2}\right)\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \end{array}\]

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 3 regimes

  2. if (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)) < -inf.0

    1. Initial program 62.4

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

    3. Applied div-inv62.4

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

    4. Applied associate-*l*62.4

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

    5. Taylor expanded around 0 46.8

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

    if -inf.0 < (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)) < 0.9730656778675094

    1. Initial program 0.4

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

    2. Taylor expanded around -inf 0.4

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

    if 0.9730656778675094 < (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))

    1. Initial program 43.9

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

    2. Taylor expanded around -inf 43.9

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

    4. Applied add-sqr-sqrt43.9

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

    5. Applied sqrt-prod44.1

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

    6. Taylor expanded around 0 48.8

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} = -\infty:\\ \;\;\;\;\sin ky \cdot \left(\frac{1}{\left(ky + \frac{1}{12} \cdot \left(ky \cdot {kx}^{2}\right)\right) - {ky}^{3} \cdot \frac{1}{6}} \cdot \sin th\right)\\ \mathbf{elif}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9730656778675094:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + \frac{1}{12} \cdot \left(ky \cdot {kx}^{2}\right)\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \end{array}\]

Runtime

Time bar (total: 26.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes3.73.52.51.220.1%
herbie shell --seed 2018295 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))