Average Error: 0.0 → 0.0
Time: 18.5s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(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: 17.6s)Debug log

sample230.0ms

Algorithm
halfpoints

simplify193.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

192.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 2 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

6.0ms
(/ (cos y) (exp x))
0.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

series158.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

116.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
42.0ms
(/ (cos y) (exp x))

simplify351.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

118.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
106.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
30.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune615.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 242
Calls

4 calls. Slowest were:

8.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
4.0ms
(/ (cos y) (sqrt (exp x)))
1.0ms
(sqrt (exp x))

series205.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(fma (cos y) (exp x) (/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x))))
70.0ms
(/ (cos y) (sqrt (exp x)))
35.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
20.0ms
(sqrt (exp x))

simplify6.4s

Counts
358 → 254
Calls

358 calls. Slowest were:

405.0ms
(/ (* (* (/ (cos y) (sqrt (exp x))) (/ (cos y) (sqrt (exp x)))) (/ (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
349.0ms
(/ (/ 1 (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) (sqrt (sqrt (exp x))))
255.0ms
(/ (/ (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))

prune2.7s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes72.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample6.5s

Algorithm
halfpoints