Average Error: 0.0 → 0.0
Time: 19.4s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\sqrt{e^{x}} \cdot \left(\sqrt{e^{x}} \cdot \cos y\right) + \frac{\cos y}{e^{x}}}{2}\]

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. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019018 
(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: 18.7s)Debug log

sample178.0ms

Algorithm
halfpoints
Results

simplify163.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
163.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune29.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize36.0ms

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))

rewrite31.0ms

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
15.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
8.0ms
(* (cos y) (exp x))
6.0ms
(/ (cos y) (exp x))

series193.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
90.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
71.0ms
(/ (cos y) (exp x))
32.0ms
(* (cos y) (exp x))

simplify432.0ms

Counts
45 → 65
Calls
45 calls:
Slowest
129.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
59.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
48.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
43.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
31.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune584.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
17×add-cube-cbrt
17×*-un-lft-identity
17×add-sqr-sqrt
13×add-cbrt-cube
12×add-exp-log
12×associate-*r*
11×pow1
9×times-frac
6×add-log-exp
6×sqrt-prod
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
4×associate-*l*
3×associate-/l*
3×associate-/r*
2×*-commutative
1×div-inv
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
4 → 79
Calls
4 calls:
Slowest
14.0ms
(+ (/ (cos y) (exp x)) (* (* (cos y) (sqrt (exp x))) (sqrt (exp x))))
13.0ms
(* (* (cos y) (sqrt (exp x))) (sqrt (exp x)))
4.0ms
(* (cos y) (sqrt (exp x)))
4.0ms
(/ (cos y) (exp x))

series198.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
73.0ms
(+ (/ (cos y) (exp x)) (* (* (cos y) (sqrt (exp x))) (sqrt (exp x))))
50.0ms
(/ (cos y) (exp x))
39.0ms
(* (cos y) (sqrt (exp x)))
36.0ms
(* (* (cos y) (sqrt (exp x))) (sqrt (exp x)))

simplify1.3s

Counts
62 → 91
Calls
62 calls:
Slowest
461.0ms
(* (* (* (* (cos y) (sqrt (exp x))) (* (cos y) (sqrt (exp x)))) (* (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
292.0ms
(* (* (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
144.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
61.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
60.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))

prune996.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize12.0ms

Local error

Found 2 expressions with local error:

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

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×associate-*r*
4×add-cbrt-cube
4×pow1
4×add-sqr-sqrt
3×associate-*l*
2×distribute-lft-out
2×flip-+
2×associate-*r/
2×flip3-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×distribute-lft-in
1×sum-log
1×+-commutative
1×distribute-rgt-in
Counts
2 → 34
Calls
2 calls:
Slowest
14.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
7.0ms
(+ (exp (- x)) (exp x))

series85.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
60.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
25.0ms
(+ (exp (- x)) (exp x))

simplify697.0ms

Counts
21 → 40
Calls
21 calls:
Slowest
188.0ms
(+ (* (exp (- x)) (exp (- x))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp x))))
180.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
139.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (+ (exp (- x)) (exp x)) (+ (exp (- x)) (exp x))) (+ (exp (- x)) (exp x))))
33.0ms
(* (+ (exp (* -1 x)) (exp x)) (cos y))
32.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune419.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize11.0ms

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

rewrite39.0ms

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
19.0ms
(+ (/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x))) (* (cos y) (exp x)))
8.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
4.0ms
(/ (cos y) (sqrt (exp x)))
4.0ms
(* (cos y) (exp x))

series232.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
93.0ms
(+ (/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x))) (* (cos y) (exp x)))
48.0ms
(/ (cos y) (sqrt (exp x)))
48.0ms
(* (cos y) (exp x))
43.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))

simplify5.4s

Counts
363 → 254
Calls
363 calls:
Slowest
372.0ms
(/ (* (* (/ (cos y) (sqrt (exp x))) (/ (cos y) (sqrt (exp x)))) (/ (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
203.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) (sqrt 1))
191.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) 1)
159.0ms
(/ (/ (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
145.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (sqrt 1)) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x)))))

prune2.4s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes79.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample5.1s

Algorithm
halfpoints
Results