Average Error: 0.5 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a2 \cdot a2\right)\right) + \frac{a1 \cdot a1}{\sqrt{2}} \cdot \cos th\]

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqr-sqrt0.5

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

  6. Applied associate-/r*0.5

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

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

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

  9. Applied div-inv0.5

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

  10. Applied times-frac0.5

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

  11. Applied associate-*l*0.5

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

  12. Simplified0.5

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

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

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

  15. Applied times-frac0.5

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

  16. Simplified0.5

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

  17. Final simplification0.5

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))