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

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

    \[\leadsto \frac{\cos th}{\sqrt{2}} \cdot (a1 \cdot a1 + \left(a2 \cdot a2\right))_*\]
  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 fma-udef0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Simplified0.5

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

  11. Applied add-sqr-sqrt0.5

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

  12. Applied associate-/l*0.5

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

  13. Final simplification0.5

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018234 +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))))