Average Error: 0.5 → 0.5
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}} + \left(\frac{\frac{\cos th}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right) \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

  1. Initial program 0.5

    \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
  2. Using strategy rm

  3. Applied associate-*l/0.5

    \[\leadsto \color{blue}{\frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}} + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \frac{\cos th}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

  6. Applied associate-/r*0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\frac{\cos th}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \frac{\frac{\cos th}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}} \cdot \left(a2 \cdot a2\right)\]

  9. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \frac{\frac{\color{blue}{1 \cdot \cos th}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

  10. Applied times-frac0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \frac{\color{blue}{\frac{1}{\sqrt[3]{\sqrt{2}}} \cdot \frac{\cos th}{\sqrt[3]{\sqrt{2}}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\]

  11. Applied times-frac0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \frac{\frac{\cos th}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}\right)} \cdot \left(a2 \cdot a2\right)\]

  12. Applied associate-*l*0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(\frac{\frac{\cos th}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right)}\]

  13. Final simplification0.5

    \[\leadsto \frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}} + \left(\frac{\frac{\cos th}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right) \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))