Average Error: 0.0 → 0.0
Time: 8.3s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

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 2019007 
(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: 7.8s)Debug log

sample274.0ms

Algorithm
halfpoints

simplify143.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

143.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize42.0ms

Local error

Found 3 expressions with local error:

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

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
3 → 56
Calls

3 calls. Slowest were:

16.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
7.0ms
(* (cos y) (exp x))
6.0ms
(/ (cos y) (exp x))

series180.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

84.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
48.0ms
(* (cos y) (exp x))
47.0ms
(/ (cos y) (exp x))

simplify403.0ms

Counts
45 → 65
Calls

45 calls. Slowest were:

141.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
54.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
49.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune689.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize19.0ms

Local error

Found 2 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

11.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
5.0ms
(+ (exp (- x)) (exp x))

series110.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

76.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
33.0ms
(+ (exp (- x)) (exp x))

simplify594.0ms

Counts
21 → 40
Calls

21 calls. Slowest were:

155.0ms
(+ (* (exp (- x)) (exp (- x))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp x))))
137.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
128.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (+ (exp (- x)) (exp x)) (+ (exp (- x)) (exp x))) (+ (exp (- x)) (exp x))))

prune355.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes49.0ms

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