Average Error: 0.0 → 0.0
Time: 10.4s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\cos y \cdot e^{x} + \frac{1}{e^{x}} \cdot \cos y}{2}\]

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. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot \frac{1}{e^{x}}} + \cos y \cdot e^{x}}{2}\]

  5. Final simplification0.0

    \[\leadsto \frac{\cos y \cdot e^{x} + \frac{1}{e^{x}} \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 2019005 
(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: 9.8s)Debug log

sample136.0ms

Algorithm
halfpoints

simplify81.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

81.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune13.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 3 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
3 → 56
Calls

3 calls. Slowest were:

8.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
8.0ms
(* (cos y) (exp x))
3.0ms
(/ (cos y) (exp x))

series160.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

92.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
35.0ms
(* (cos y) (exp x))
32.0ms
(/ (cos y) (exp x))

simplify469.0ms

Counts
45 → 65
Calls

45 calls. Slowest were:

115.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
85.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
80.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))

prune653.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize11.0ms

Local error

Found 4 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 66
Calls

4 calls. Slowest were:

7.0ms
(+ (* (cos y) (/ 1 (exp x))) (* (cos y) (exp x)))
4.0ms
(* (cos y) (/ 1 (exp x)))
4.0ms
(* (cos y) (exp x))

series129.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

63.0ms
(+ (* (cos y) (/ 1 (exp x))) (* (cos y) (exp x)))
29.0ms
(* (cos y) (/ 1 (exp x)))
28.0ms
(* (cos y) (exp x))
8.0ms
(/ 1 (exp x))

simplify548.0ms

Counts
39 → 78
Calls

39 calls. Slowest were:

281.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (/ 1 (exp x)) (/ 1 (exp x))) (/ 1 (exp x))))
59.0ms
(* (exp (* (cos y) (/ 1 (exp x)))) (exp (* (cos y) (exp x))))
43.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune745.0ms

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

10.0ms
(* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x))))
2.0ms
(sqrt (/ 1 (exp x)))
1.0ms
(sqrt (/ 1 (exp x)))

series60.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

16.0ms
(sqrt (/ 1 (exp x)))
15.0ms
(sqrt (/ 1 (exp x)))
14.0ms
(/ 1 (exp x))
14.0ms
(* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x))))

simplify461.0ms

Counts
56 → 91
Calls

56 calls. Slowest were:

115.0ms
(* (* (* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x)))) (sqrt (/ 1 (exp x)))) (* (* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x)))) (sqrt (/ 1 (exp x)))))
53.0ms
(- (+ (* 1/8 (pow x 2)) 1) (* 1/2 x))
36.0ms
(* (/ 1 (exp x)) (/ 1 (exp x)))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes69.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample5.0s

Algorithm
halfpoints