Average Error: 0.0 → 0.0
Time: 25.3s
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 1137770271 
(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: 24.6s)Debug log

sample305.0ms

Algorithm
halfpoints
Results

simplify187.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
187.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize49.0ms

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))

rewrite44.0ms

Algorithm
rewrite-expression-head
Rules
13×add-cube-cbrt
13×*-un-lft-identity
13×add-sqr-sqrt
9×times-frac
7×add-cbrt-cube
5×add-log-exp
5×add-exp-log
5×pow1
3×associate-/l*
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×div-inv
1×distribute-lft-out
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×div-exp
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
3 → 56
Calls
3 calls:
Slowest
20.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
13.0ms
(/ (cos y) (exp x))
9.0ms
(* (cos y) (exp x))

series263.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
137.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
67.0ms
(/ (cos y) (exp x))
59.0ms
(* (cos y) (exp x))

simplify667.0ms

Counts
45 → 65
Calls
45 calls:
Slowest
184.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
101.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
98.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
52.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
38.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune1.1s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize40.0ms

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

rewrite90.0ms

Algorithm
rewrite-expression-head
Rules
276×times-frac
158×add-cube-cbrt
158×*-un-lft-identity
158×add-sqr-sqrt
144×sqrt-prod
25×associate-/l*
13×add-cbrt-cube
12×add-exp-log
12×associate-/r*
9×div-inv
6×add-log-exp
6×pow1
4×div-exp
4×cbrt-undiv
3×associate-*l*
3×associate-*r*
2×frac-2neg
2×clear-num
1×associate-/l/
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×flip3-+
1×sum-log
1×+-commutative
Counts
4 → 242
Calls
4 calls:
Slowest
43.0ms
(+ (/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x))) (* (cos y) (exp x)))
18.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
9.0ms
(* (cos y) (exp x))
9.0ms
(/ (cos y) (sqrt (exp x)))

series348.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
131.0ms
(+ (/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x))) (* (cos y) (exp x)))
88.0ms
(/ (cos y) (sqrt (exp x)))
68.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
60.0ms
(* (cos y) (exp x))

simplify8.7s

Counts
363 → 254
Calls
363 calls:
Slowest
625.0ms
(/ (* (* (/ (cos y) (sqrt (exp x))) (/ (cos y) (sqrt (exp x)))) (/ (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
297.0ms
(/ (/ (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
278.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) 1)
261.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) (sqrt 1))
179.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (sqrt 1)) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x)))))

prune5.2s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes113.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample7.6s

Algorithm
halfpoints
Results