Average Error: 0.0 → 0.0
Time: 34.4s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{e^{x} \cdot \cos y + \frac{\cos y}{e^{x}}}{2}\]

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 2019021 
(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: 33.4s)Debug log

sample154.0ms

Algorithm
halfpoints
Results

simplify88.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
88.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 3 expressions with local error:

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

rewrite22.0ms

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
9.0ms
(* (cos y) (exp x))
8.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
3.0ms
(/ (cos y) (exp x))

series122.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
62.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
32.0ms
(/ (cos y) (exp x))
27.0ms
(* (cos y) (exp x))

simplify316.0ms

Counts
45 → 65
Calls
45 calls:
Slowest
84.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
48.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
44.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
26.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
18.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune479.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.4b
(cbrt (pow (/ (cos y) (exp x)) 3))
0.3b
(pow (/ (cos y) (exp x)) 3)
0.0b
(/ (cos y) (exp x))
0.0b
(* (cos y) (exp x))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
45×times-frac
43×add-cube-cbrt
43×*-un-lft-identity
43×add-sqr-sqrt
31×cbrt-prod
26×cube-prod
26×unpow-prod-down
11×add-cbrt-cube
8×add-exp-log
7×pow1
5×div-inv
4×add-log-exp
3×associate-/l*
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×rem-cube-cbrt
2×pow-exp
2×div-exp
2×unpow3
2×cbrt-undiv
2×cube-mult
2×cube-div
1×cbrt-unprod
1×rem-cbrt-cube
1×*-commutative
1×prod-exp
1×pow-prod-down
1×pow1/3
1×frac-2neg
1×pow-pow
1×clear-num
1×cbrt-div
1×pow-to-exp
Counts
4 → 127
Calls
4 calls:
Slowest
3.0ms
(* (cos y) (exp x))
3.0ms
(/ (cos y) (exp x))
2.0ms
(pow (/ (cos y) (exp x)) 3)
2.0ms
(cbrt (pow (/ (cos y) (exp x)) 3))

series146.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
43.0ms
(pow (/ (cos y) (exp x)) 3)
42.0ms
(* (cos y) (exp x))
31.0ms
(cbrt (pow (/ (cos y) (exp x)) 3))
29.0ms
(/ (cos y) (exp x))

simplify10.9s

Counts
110 → 139
Calls
110 calls:
Slowest
1.1s
(pow (/ 1 (* (cbrt (exp x)) (cbrt (exp x)))) 3)
1.1s
(pow (/ 1 (* (cbrt (exp x)) (cbrt (exp x)))) 3)
818.0ms
(pow (/ (* (cbrt (cos y)) (cbrt (cos y))) 1) 3)
727.0ms
(cbrt (pow (/ (* (cbrt (cos y)) (cbrt (cos y))) 1) 3))
639.0ms
(pow (/ (* (cbrt (cos y)) (cbrt (cos y))) 1) 3)

prune1.7s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize17.0ms

Local error

Found 4 expressions with local error:

0.4b
(cbrt (pow (/ (cos y) (exp x)) 3))
0.3b
(pow (/ (cos y) (exp x)) 3)
0.3b
(cbrt (* (cos y) (exp x)))
0.3b
(cbrt (* (cos y) (exp x)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
36×times-frac
33×add-cube-cbrt
33×*-un-lft-identity
33×add-sqr-sqrt
33×cbrt-prod
26×cube-prod
26×unpow-prod-down
7×add-cbrt-cube
6×add-exp-log
5×pow1
4×add-log-exp
4×div-inv
3×pow1/3
2×rem-cube-cbrt
2×pow-exp
2×unpow3
2×cube-mult
2×cube-div
1×rem-cbrt-cube
1×div-exp
1×pow-pow
1×cbrt-div
1×cbrt-undiv
1×pow-to-exp
Counts
4 → 101
Calls
4 calls:
Slowest
4.0ms
(cbrt (* (cos y) (exp x)))
3.0ms
(cbrt (* (cos y) (exp x)))
3.0ms
(pow (/ (cos y) (exp x)) 3)
2.0ms
(cbrt (pow (/ (cos y) (exp x)) 3))

series287.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
87.0ms
(cbrt (* (cos y) (exp x)))
83.0ms
(cbrt (* (cos y) (exp x)))
65.0ms
(pow (/ (cos y) (exp x)) 3)
53.0ms
(cbrt (pow (/ (cos y) (exp x)) 3))

simplify10.8s

Counts
79 → 113
Calls
79 calls:
Slowest
999.0ms
(pow (/ 1 (* (cbrt (exp x)) (cbrt (exp x)))) 3)
954.0ms
(pow (/ 1 (* (cbrt (exp x)) (cbrt (exp x)))) 3)
753.0ms
(pow (/ (* (cbrt (cos y)) (cbrt (cos y))) 1) 3)
685.0ms
(pow (/ (* (cbrt (cos y)) (cbrt (cos y))) 1) 3)
481.0ms
(cbrt (pow (/ (* (cbrt (cos y)) (cbrt (cos y))) (sqrt (exp x))) 3))

prune1.5s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes61.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample6.7s

Algorithm
halfpoints
Results