(+ (pow (- (* (* (pow 10 4) x) y) 1) 2) (pow (- (+ (exp (- x)) (exp (- y))) 1.0001) 2))

Average Error: 0.3 → 0.3

Time: 2.7s

Precision: binary64

Math

\[{\left(\left({10}^{4} \cdot x\right) \cdot y - 1\right)}^{2} + {\left(\left(e^{-x} + e^{-y}\right) - 1.00009999999999999\right)}^{2}\]

↓

\[{\left(\left({10}^{4} \cdot x\right) \cdot y - 1\right)}^{2} + {\left(\left(e^{-x} + e^{-y}\right) - 1.00009999999999999\right)}^{2}\]

Error

Bits error versus x

Bits error versus y

Derivation

1. Initial program 0.3

   \[{\left(\left({10}^{4} \cdot x\right) \cdot y - 1\right)}^{2} + {\left(\left(e^{-x} + e^{-y}\right) - 1.00009999999999999\right)}^{2}\]

2. Final simplification 0.3

   \[\leadsto {\left(\left({10}^{4} \cdot x\right) \cdot y - 1\right)}^{2} + {\left(\left(e^{-x} + e^{-y}\right) - 1.00009999999999999\right)}^{2}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x y)
  :name "(+ (pow (- (* (* (pow 10 4) x) y) 1) 2) (pow (- (+ (exp (- x)) (exp (- y))) 1.0001) 2))"
  :precision binary64
  (+ (pow (- (* (* (pow 10.0 4.0) x) y) 1.0) 2.0) (pow (- (+ (exp (neg x)) (exp (neg y))) 1.0001) 2.0)))