Average Error: 0.5 → 0.4
Time: 2.4m
Precision: 64
Internal Precision: 576
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{a1}{\sqrt{2}} \cdot \cos th\right) \cdot a1 + \frac{a2}{\sqrt{2}} \cdot \left(\cos th \cdot a2\right)\]

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. Applied simplify0.5

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

  4. Applied add-cube-cbrt0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied fma-udef0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Applied simplify0.5

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

  10. Applied simplify0.4

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

  12. Applied associate-*r*0.4

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018198 +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))))