Average Error: 0.5 → 0.5
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \left(\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}\right)\]

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. Initial simplification0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-/r*0.5

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

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

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

  8. Applied sqrt-prod0.5

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

  9. Applied div-inv0.6

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

  10. Applied times-frac0.5

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

  11. Applied associate-*l*0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018248 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))