Average Error: 0.5 → 0.5
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\sqrt{a1^2 + a2^2}^* \cdot \cos th}{\sqrt{2}} \cdot \sqrt{(a1 \cdot a1 + \left(a2 \cdot a2\right))_*}\]

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. 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-sqr-sqrt0.5

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

  5. Applied associate-*r*0.5

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

  6. Applied simplify0.5

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018206 +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))))