Average Error: 0.0 → 0.0
Time: 13.3s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\cos y \cdot e^{x} + \frac{1}{e^{x}} \cdot \cos y}{2}\]

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. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot \frac{1}{e^{x}}} + \cos y \cdot e^{x}}{2}\]

  5. Final simplification0.0

    \[\leadsto \frac{\cos y \cdot e^{x} + \frac{1}{e^{x}} \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 323378853 
(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: 12.7s)Debug log

sample290.0ms

Algorithm
halfpoints
Results

simplify191.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
191.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune33.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize48.0ms

Local error

Found 2 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
12×add-cube-cbrt
12×*-un-lft-identity
12×add-sqr-sqrt
9×times-frac
6×add-cbrt-cube
4×add-exp-log
4×pow1
3×associate-/l*
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×add-log-exp
1×div-inv
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 44
Calls
2 calls:
Slowest
9.0ms
(* (cos y) (exp x))
7.0ms
(/ (cos y) (exp x))

series122.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
62.0ms
(/ (cos y) (exp x))
59.0ms
(* (cos y) (exp x))

simplify596.0ms

Counts
39 → 50
Calls
39 calls:
Slowest
184.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
102.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
93.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
53.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
34.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))

prune800.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

Found 3 expressions with local error:

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

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×pow1
8×add-sqr-sqrt
7×add-exp-log
7×associate-*r*
7×add-cbrt-cube
6×associate-*l*
3×add-log-exp
3×prod-exp
3×associate-/r*
2×div-inv
2×cbrt-unprod
2×*-commutative
2×pow-prod-down
2×rec-exp
1×inv-pow
1×pow-flip
1×un-div-inv
1×associate-*r/
1×frac-2neg
1×clear-num
Counts
3 → 54
Calls
3 calls:
Slowest
9.0ms
(* (cos y) (/ 1 (exp x)))
9.0ms
(* (cos y) (exp x))
2.0ms
(/ 1 (exp x))

series140.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
61.0ms
(* (cos y) (exp x))
61.0ms
(* (cos y) (/ 1 (exp x)))
17.0ms
(/ 1 (exp x))

simplify1.0s

Counts
33 → 63
Calls
33 calls:
Slowest
600.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (/ 1 (exp x)) (/ 1 (exp x))) (/ 1 (exp x))))
102.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
95.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
55.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
55.0ms
(- (+ (* 1/2 (pow x 2)) 1) x)

prune1.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes105.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample7.8s

Algorithm
halfpoints
Results