Average Error: 0.0 → 0.0
Time: 11.6s
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. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018362 
(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: 4.9s)Debug log

start211.0ms

Algorithm
halfpoints

setup185.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

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))

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))

series163.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

77.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
52.0ms
(/ (cos y) (exp x))
33.0ms
(* (cos y) (exp x))

simplify483.0ms

Counts
45 → 65
Calls

45 calls. Slowest were:

160.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
82.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
44.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune654.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 2 expressions with local error:

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

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
2 → 34
Calls

2 calls. Slowest were:

21.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
11.0ms
(+ (exp (- x)) (exp x))

series105.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

63.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
41.0ms
(+ (exp (- x)) (exp x))

simplify544.0ms

Counts
21 → 40
Calls

21 calls. Slowest were:

169.0ms
(+ (* (exp (- x)) (exp (- x))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp x))))
114.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
88.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (+ (exp (- x)) (exp x)) (+ (exp (- x)) (exp x))) (+ (exp (- x)) (exp x))))

prune463.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize16.0ms

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)))

rewrite25.0ms

Algorithm
rewrite-expression-head
Counts
3 → 49
Calls

3 calls. Slowest were:

14.0ms
(+ (* (cos y) (exp (- x))) (* (cos y) (exp x)))
6.0ms
(* (cos y) (exp (- x)))
3.0ms
(* (cos y) (exp x))

series183.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

85.0ms
(+ (* (cos y) (exp (- x))) (* (cos y) (exp x)))
51.0ms
(* (cos y) (exp (- x)))
47.0ms
(* (cos y) (exp x))

simplify826.0ms

Counts
31 → 58
Calls

31 calls. Slowest were:

234.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp (- x)) (exp (- x))) (exp (- x))))
142.0ms
(* (exp (* (cos y) (exp (- x)))) (exp (* (cos y) (exp x))))
118.0ms
(+ (* (cos y) (exp (- x))) (* (exp x) (cos y)))

prune863.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes70.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms