Average Error: 0.0 → 0.0
Time: 10.1s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\left(e^{x} + e^{-x}\right) \cdot \cos y}{2}\]

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. Applied distribute-lft-out0.0

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

  6. Simplified0.0

    \[\leadsto \frac{\cos y \cdot \color{blue}{\left(e^{-x} + e^{x}\right)}}{2}\]

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019008 
(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: 9.5s)Debug log

sample136.0ms

Algorithm
halfpoints

simplify79.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

79.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

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)))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
3 → 56
Calls

3 calls. Slowest were:

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

series139.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

63.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
41.0ms
(* (cos y) (exp x))
35.0ms
(/ (cos y) (exp x))

simplify290.0ms

Counts
45 → 65
Calls

45 calls. Slowest were:

88.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
41.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
40.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))

prune429.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize14.0ms

Local error

Found 2 expressions with local error:

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

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

19.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
11.0ms
(+ (exp (- x)) (exp x))

series126.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

84.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
42.0ms
(+ (exp (- x)) (exp x))

simplify548.0ms

Counts
21 → 40
Calls

21 calls. Slowest were:

188.0ms
(+ (* (exp (- x)) (exp (- x))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp x))))
145.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
71.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (+ (exp (- x)) (exp x)) (+ (exp (- x)) (exp x))) (+ (exp (- x)) (exp x))))

prune309.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

11.0ms
(* (* (cos y) (sqrt (exp x))) (sqrt (exp x)))
4.0ms
(* (cos y) (sqrt (exp x)))
3.0ms
(/ (cos y) (exp x))

series142.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

55.0ms
(* (cos y) (sqrt (exp x)))
41.0ms
(/ (cos y) (exp x))
28.0ms
(* (* (cos y) (sqrt (exp x))) (sqrt (exp x)))
18.0ms
(sqrt (exp x))

simplify1.3s

Counts
65 → 92
Calls

65 calls. Slowest were:

421.0ms
(* (* (* (* (cos y) (sqrt (exp x))) (* (cos y) (sqrt (exp x)))) (* (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
321.0ms
(* (* (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
150.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune983.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes57.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample4.9s

Algorithm
halfpoints