Average Error: 0.0 → 0.0
Time: 9.0s
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 2019016 
(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: 8.5s)Debug log

sample162.0ms

Algorithm
halfpoints
Results

simplify93.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
93.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 2 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×times-frac
4×add-log-exp
4×add-cbrt-cube
3×associate-/l*
3×add-exp-log
3×associate-/r*
2×div-inv
2×pow1
1×distribute-lft-out
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
2 → 39
Calls
2 calls:
Slowest
8.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
3.0ms
(/ (cos y) (exp x))

series107.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
75.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
31.0ms
(/ (cos y) (exp x))

simplify264.0ms

Counts
34 → 45
Calls
34 calls:
Slowest
130.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
41.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
17.0ms
(- (+ (pow x 2) 2) (pow y 2))
14.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))
9.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) 1)

prune409.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.0b
(* (/ 1 (sqrt (exp x))) (/ (cos y) (sqrt (exp x))))
0.0b
(/ (cos y) (sqrt (exp x)))
0.0b
(/ 1 (sqrt (exp x)))
0.0b
(sqrt (exp x))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
37×add-sqr-sqrt
36×add-cube-cbrt
36×times-frac
36×*-un-lft-identity
27×sqrt-prod
22×associate-*r*
17×add-exp-log
12×associate-/r*
11×add-cbrt-cube
7×pow1
4×add-log-exp
4×div-inv
4×prod-exp
4×associate-*l*
3×associate-/l*
3×div-exp
3×rec-exp
2×pow-flip
2×cbrt-unprod
2×pow1/2
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×inv-pow
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×rem-sqrt-square
1×frac-times
Counts
4 → 115
Calls
4 calls:
Slowest
6.0ms
(* (/ 1 (sqrt (exp x))) (/ (cos y) (sqrt (exp x))))
3.0ms
(/ (cos y) (sqrt (exp x)))
1.0ms
(/ 1 (sqrt (exp x)))
1.0ms
(sqrt (exp x))

series120.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
53.0ms
(/ (cos y) (sqrt (exp x)))
37.0ms
(* (/ 1 (sqrt (exp x))) (/ (cos y) (sqrt (exp x))))
15.0ms
(/ 1 (sqrt (exp x)))
14.0ms
(sqrt (exp x))

simplify1.2s

Counts
105 → 127
Calls
105 calls:
Slowest
277.0ms
(* (* (* (/ 1 (sqrt (exp x))) (/ 1 (sqrt (exp x)))) (/ 1 (sqrt (exp x)))) (* (* (/ (cos y) (sqrt (exp x))) (/ (cos y) (sqrt (exp x)))) (/ (cos y) (sqrt (exp x)))))
97.0ms
(+ (log (/ 1 (sqrt (exp x)))) (- (log (cos y)) (log (sqrt (exp x)))))
92.0ms
(+ (log (/ 1 (sqrt (exp x)))) (log (/ (cos y) (sqrt (exp x)))))
79.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
74.0ms
(* (* (* (/ 1 (sqrt (exp x))) (/ 1 (sqrt (exp x)))) (/ 1 (sqrt (exp x)))) (/ (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes41.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch1.0ms

end0.0ms

sample4.8s

Algorithm
halfpoints
Results