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

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. Initial simplification0.5

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

  4. Applied associate-*l/0.5

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

  5. Final simplification0.5

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

Runtime

Time bar (total: 1.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.50.50.00.40%
herbie shell --seed 2018296 +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))))