Average Error: 0.0 → 0.0
Time: 10.8s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{1}{e^{x}} \cdot \cos y\right))_*}{2}\]

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. Using strategy rm

  4. Applied div-inv0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019005 +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: 10.0s)Debug log

sample175.0ms

Algorithm
halfpoints

simplify124.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

124.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune28.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize31.0ms

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)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

3.0ms
(/ (cos y) (exp x))
0.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

series121.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

85.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
35.0ms
(/ (cos y) (exp x))

simplify316.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

100.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
89.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
34.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune445.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize15.0ms

Local error

Found 3 expressions with local error:

0.0b
(/ 1 (exp x))
0.0b
(fma (cos y) (exp x) (* (cos y) (/ 1 (exp x))))
0.0b
(* (cos y) (/ 1 (exp x)))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
3 → 51
Calls

3 calls. Slowest were:

4.0ms
(* (cos y) (/ 1 (exp x)))
1.0ms
(/ 1 (exp x))
0.0ms
(fma (cos y) (exp x) (* (cos y) (/ 1 (exp x))))

series138.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

94.0ms
(fma (cos y) (exp x) (* (cos y) (/ 1 (exp x))))
35.0ms
(* (cos y) (/ 1 (exp x)))
8.0ms
(/ 1 (exp x))

simplify665.0ms

Counts
25 → 60
Calls

25 calls. Slowest were:

421.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (/ 1 (exp x)) (/ 1 (exp x))) (/ 1 (exp x))))
86.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
51.0ms
(- (+ (* 1/2 (pow x 2)) 1) x)

prune615.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.0b
(* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x))))
0.0b
(sqrt (/ 1 (exp x)))
0.0b
(sqrt (/ 1 (exp x)))
0.0b
(/ 1 (exp x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

5.0ms
(* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x))))
1.0ms
(sqrt (/ 1 (exp x)))
1.0ms
(sqrt (/ 1 (exp x)))

series55.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

19.0ms
(sqrt (/ 1 (exp x)))
15.0ms
(sqrt (/ 1 (exp x)))
13.0ms
(/ 1 (exp x))
8.0ms
(* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x))))

simplify497.0ms

Counts
56 → 99
Calls

56 calls. Slowest were:

143.0ms
(* (* (* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x)))) (sqrt (/ 1 (exp x)))) (* (* (sqrt (/ 1 (exp x))) (sqrt (/ 1 (exp x)))) (sqrt (/ 1 (exp x)))))
57.0ms
(- (+ (* 1/8 (pow x 2)) 1) (* 1/2 x))
50.0ms
(- (+ (* 1/8 (pow x 2)) 1) (* 1/2 x))

prune1.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes52.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample5.4s

Algorithm
halfpoints