Average Error: 0.0 → 0.0
Time: 22.4s
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 2019002 
(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: 21.5s)Debug log

sample193.0ms

Algorithm
halfpoints

simplify141.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

140.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune17.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

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:

10.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
6.0ms
(/ (cos y) (exp x))
3.0ms
(* (cos y) (exp x))

series152.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

79.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
43.0ms
(/ (cos y) (exp x))
31.0ms
(* (cos y) (exp x))

simplify474.0ms

Counts
45 → 65
Calls

45 calls. Slowest were:

142.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
69.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
51.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune596.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))
0.0b
(/ (cos y) (exp x))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

6.0ms
(/ (cos y) (exp x))
1.0ms
(cbrt (exp x))
1.0ms
(cbrt (exp x))

series412.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

122.0ms
(cbrt (exp x))
120.0ms
(cbrt (exp x))
117.0ms
(cbrt (exp x))
52.0ms
(/ (cos y) (exp x))

simplify403.0ms

Counts
46 → 72
Calls

46 calls. Slowest were:

145.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
41.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
30.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))

prune590.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize52.0ms

Local error

Found 4 expressions with local error:

18.1b
(- (* (* (cos y) (exp x)) (* (cos y) (exp x))) (* (/ (cos y) (exp x)) (* (cos y) (exp x))))
0.4b
(pow (/ (cos y) (exp x)) 3)
0.4b
(pow (* (cos y) (exp x)) 3)
0.3b
(* (/ (cos y) (exp x)) (/ (cos y) (exp x)))

rewrite53.0ms

Algorithm
rewrite-expression-head
Counts
4 → 122
Calls

4 calls. Slowest were:

29.0ms
(- (* (* (cos y) (exp x)) (* (cos y) (exp x))) (* (/ (cos y) (exp x)) (* (cos y) (exp x))))
12.0ms
(* (/ (cos y) (exp x)) (/ (cos y) (exp x)))
5.0ms
(pow (* (cos y) (exp x)) 3)

series215.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

94.0ms
(- (* (* (cos y) (exp x)) (* (cos y) (exp x))) (* (/ (cos y) (exp x)) (* (cos y) (exp x))))
41.0ms
(pow (/ (cos y) (exp x)) 3)
41.0ms
(pow (* (cos y) (exp x)) 3)
39.0ms
(* (/ (cos y) (exp x)) (/ (cos y) (exp x)))

simplify9.7s

Counts
96 → 134
Calls

96 calls. Slowest were:

1.1s
(pow (/ 1 (* (cbrt (exp x)) (cbrt (exp x)))) 3)
979.0ms
(* (* (* (/ (cos y) (exp x)) (/ (cos y) (exp x))) (/ (cos y) (exp x))) (* (* (/ (cos y) (exp x)) (/ (cos y) (exp x))) (/ (cos y) (exp x))))
978.0ms
(pow (/ 1 (* (cbrt (exp x)) (cbrt (exp x)))) 3)

prune2.1s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes106.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample6.1s

Algorithm
halfpoints