Average Error: 0.5 → 0.4
Time: 1.8m
Precision: 64
Internal Precision: 320
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{(a1 \cdot a1 + \left(a2 \cdot a2\right))_* \cdot \cos th}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{\sqrt{2}}}}\]

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. Initial simplification0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied add-sqr-sqrt0.5

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

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

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

  10. Applied times-frac0.5

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

  11. Applied times-frac0.6

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

  12. Applied associate-*l*0.6

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

  13. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\sqrt{2}}}} \cdot \left(\frac{\frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\right)\]
  14. Using strategy rm

  15. Applied associate-*l/0.5

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

  16. Applied frac-times0.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))