Average Error: 12.5 → 11.8
Time: 36.5s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9999999999864747:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin kx}\right)}^{2} \cdot \left(\sin kx \cdot \left(\sqrt[3]{\sqrt[3]{\sin kx}} \cdot \sqrt[3]{\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}}\right)\right) + {\left(\sin ky\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{6} \cdot \left(kx \cdot kx\right) + 1\right) \cdot \sin th\\ \end{array}\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Derivation

  1. Split input into 2 regimes

  2. if (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) < 0.9999999999864747

    1. Initial program 13.3

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

    3. Applied add-cube-cbrt13.6

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

    4. Applied unpow-prod-down13.6

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

    5. Simplified13.5

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

    7. Applied add-cube-cbrt13.5

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

    8. Applied cbrt-prod13.5

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

    if 0.9999999999864747 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))

    1. Initial program 9.1

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

    3. Applied *-un-lft-identity9.1

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

    4. Applied associate-/l*9.1

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

    5. Taylor expanded around 0 5.0

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

    6. Simplified5.0

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9999999999864747:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sqrt{{\left(\sqrt[3]{\sin kx}\right)}^{2} \cdot \left(\sin kx \cdot \left(\sqrt[3]{\sqrt[3]{\sin kx}} \cdot \sqrt[3]{\sqrt[3]{\sin kx} \cdot \sqrt[3]{\sin kx}}\right)\right) + {\left(\sin ky\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{6} \cdot \left(kx \cdot kx\right) + 1\right) \cdot \sin th\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))