(/ (exp (/ (- (+ (pow x 2) (pow y 2))) (* 2 (pow sigma 2)))) (* (* 2 PI) (pow sigma 2)))

Average Error: 21.8 → 21.8

Time: 1.7s

Precision: binary64

Math

\[\frac{e^{\frac{-\left({x}^{2} + {y}^{2}\right)}{2 \cdot {sigma}^{2}}}}{\left(2 \cdot \pi\right) \cdot {sigma}^{2}}\]

↓

\[\frac{e^{\frac{-\left({x}^{2} + {y}^{2}\right)}{2 \cdot {sigma}^{2}}}}{\left(2 \cdot \pi\right) \cdot {sigma}^{2}}\]

Error

Bits error versus x

Bits error versus y

Bits error versus sigma

Derivation

1. Initial program 21.8

   \[\frac{e^{\frac{-\left({x}^{2} + {y}^{2}\right)}{2 \cdot {sigma}^{2}}}}{\left(2 \cdot \pi\right) \cdot {sigma}^{2}}\]

2. Final simplification 21.8

   \[\leadsto \frac{e^{\frac{-\left({x}^{2} + {y}^{2}\right)}{2 \cdot {sigma}^{2}}}}{\left(2 \cdot \pi\right) \cdot {sigma}^{2}}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x y sigma)
  :name "(/ (exp (/ (- (+ (pow x 2) (pow y 2))) (* 2 (pow sigma 2)))) (* (* 2 PI) (pow sigma 2)))"
  :precision binary64
  (/ (exp (/ (neg (+ (pow x 2.0) (pow y 2.0))) (* 2.0 (pow sigma 2.0)))) (* (* 2.0 PI) (pow sigma 2.0))))