Average Error: 0.0 → 0.0
Time: 8.2s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]

  3. Final simplification0.0

    \[\leadsto \frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))

Details

Time bar (total: 7.4s)Debug log

sample220.0ms

Algorithm
halfpoints

simplify115.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
115.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 3 expressions with local error:

0.0b
(/ (cos y) (exp x))
0.0b
(* (cos y) (exp x))
0.0b
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
13×add-cube-cbrt
13×*-un-lft-identity
13×add-sqr-sqrt
9×times-frac
7×add-cbrt-cube
5×add-log-exp
5×add-exp-log
5×pow1
3×associate-/l*
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×div-inv
1×distribute-lft-out
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×div-exp
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
3 → 56
Calls
3 calls:
Slowest
15.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
3.0ms
(* (cos y) (exp x))
3.0ms
(/ (cos y) (exp x))

series186.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
88.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
53.0ms
(* (cos y) (exp x))
45.0ms
(/ (cos y) (exp x))

simplify447.0ms

Counts
45 → 65
Calls
45 calls:
Slowest
112.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
90.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
61.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
32.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
29.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))

prune636.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample5.8s

Algorithm
halfpoints