Average Error: 3.7 → 4.3
Time: 29.5s
Precision: 64
Internal Precision: 576
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin ky}{\sqrt{\sqrt[3]{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}} \cdot \left|\sqrt[3]{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}\right|} \cdot \sin th\]

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. Initial program 3.7

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

  3. Applied add-cube-cbrt4.3

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

  4. Applied sqrt-prod4.3

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

  5. Simplified4.3

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

  6. Final simplification4.3

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

Runtime

Time bar (total: 29.5s)Debug logProfile

herbie shell --seed 2018219 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))