Average Error: 43.5 → 0.7
Time: 41.6s
Precision: 64
Internal Precision: 1344
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(\frac{1}{60} \cdot {x}^{5} + \frac{1}{3} \cdot {x}^{3}\right) + 2 \cdot x}{2} \cdot \sin y i\right))\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.5

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]

  3. Final simplification0.7

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\left(\frac{1}{60} \cdot {x}^{5} + \frac{1}{3} \cdot {x}^{3}\right) + 2 \cdot x}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 1042149663 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))

Details

Time bar (total: 37.7s)Debug log

sample671.0ms

Algorithm
halfpoints
Results

simplify186.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
186.0ms
(im (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: 41.3b

localize160.0ms

Local error

Found 4 expressions with local error:

57.1b
(- (exp x) (exp (- x)))
0.1b
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
0.0b
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
0.0b
(+ (exp x) (exp (- x)))

rewrite65.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
10×add-sqr-sqrt
8×add-log-exp
8×add-cube-cbrt
8×add-exp-log
8×associate-*l*
8×add-cbrt-cube
8×pow1
6×associate-*r*
2×div-inv
2×cbrt-unprod
2×*-commutative
2×prod-exp
2×associate-*l/
2×pow-prod-down
1×difference-of-squares
1×distribute-lft-out--
1×distribute-lft-out
1×flip--
1×flip-+
1×diff-log
1×flip3--
1×flip3-+
1×sub-neg
1×sum-log
1×+-commutative
Counts
4 → 64
Calls
4 calls:
Slowest
23.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
22.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
10.0ms
(- (exp x) (exp (- x)))
9.0ms
(+ (exp x) (exp (- x)))

series386.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
146.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
106.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
83.0ms
(- (exp x) (exp (- x)))
50.0ms
(+ (exp x) (exp (- x)))

simplify3.5s

Counts
38 → 76
Calls
38 calls:
Slowest
624.0ms
(* (* (* (/ (- (exp x) (exp (- x))) 2) (/ (- (exp x) (exp (- x))) 2)) (/ (- (exp x) (exp (- x))) 2)) (* (* (sin y) (sin y)) (sin y)))
435.0ms
(* 1/2 (* (- (exp x) (exp (- x))) (sin y)))
425.0ms
(* (* (* (/ (+ (exp x) (exp (- x))) 2) (/ (+ (exp x) (exp (- x))) 2)) (/ (+ (exp x) (exp (- x))) 2)) (* (* (cos y) (cos y)) (cos y)))
411.0ms
(* 1/2 (* (- (exp x) (exp (* -1 x))) (sin y)))
302.0ms
(* 1/2 (* (+ (exp (* -1 x)) (exp x)) (cos y)))

prune1.7s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize90.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/3 (pow x 3))
0.1b
(* (/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) 2) (sin y))
0.0b
(* 1/60 (pow x 5))
0.0b
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))

rewrite56.0ms

Algorithm
rewrite-expression-head
Rules
14×associate-*r*
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
8×add-exp-log
8×associate-*l*
8×add-cbrt-cube
8×pow1
4×add-log-exp
4×*-commutative
2×div-inv
2×cbrt-unprod
2×prod-exp
2×associate-*l/
2×pow-prod-down
1×unpow3
1×cube-mult
Counts
4 → 62
Calls
4 calls:
Slowest
26.0ms
(* (/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) 2) (sin y))
23.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
3.0ms
(* 1/3 (pow x 3))
3.0ms
(* 1/60 (pow x 5))

series294.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
120.0ms
(* (/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) 2) (sin y))
100.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
41.0ms
(* 1/60 (pow x 5))
33.0ms
(* 1/3 (pow x 3))

simplify2.5s

Counts
39 → 74
Calls
39 calls:
Slowest
408.0ms
(* (* (* (/ (+ (exp x) (exp (- x))) 2) (/ (+ (exp x) (exp (- x))) 2)) (/ (+ (exp x) (exp (- x))) 2)) (* (* (cos y) (cos y)) (cos y)))
269.0ms
(* 1/2 (* (+ (exp (* -1 x)) (exp x)) (cos y)))
246.0ms
(* 1/2 (* (+ (exp x) (exp (- x))) (cos y)))
223.0ms
(* (/ (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) 2) (sin y))
205.0ms
(* (+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5)))) (sin y))

prune2.2s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

localize65.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.5b
(* (cbrt (sin y)) (cbrt (sin y)))

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
15×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
7×pow1/3
6×add-exp-log
6×associate-*l*
6×associate-*r*
4×add-log-exp
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 61
Calls
4 calls:
Slowest
17.0ms
(* (cbrt (sin y)) (cbrt (sin y)))
2.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
275.0ms
(cbrt (sin y))
258.0ms
(* (cbrt (sin y)) (cbrt (sin y)))
240.0ms
(cbrt (sin y))
240.0ms
(cbrt (sin y))

simplify538.0ms

Counts
41 → 73
Calls
41 calls:
Slowest
135.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
111.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
107.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
77.0ms
(- (+ (pow y 2/3) (* 1/405 (pow (pow y 14) 1/3))) (* 1/9 (pow (pow y 8) 1/3)))
6.0ms
(pow (sin y) 1/3)

prune3.0s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

localize45.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
12×cbrt-prod
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
4×add-log-exp
4×add-exp-log
4×pow1/3
4×add-cbrt-cube
4×pow1
Counts
4 → 44
Calls
4 calls:
Slowest
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
264.0ms
(cbrt (sin y))
260.0ms
(cbrt (sin y))
252.0ms
(cbrt (sin y))
241.0ms
(cbrt (sin y))

simplify467.0ms

Counts
24 → 56
Calls
24 calls:
Slowest
104.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
103.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
102.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
101.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
3.0ms
(pow (sin y) 1/3)

prune2.1s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.0b

regimes431.0ms

Accuracy

0% (0.6b remaining)

Error of 0.7b against oracle of 0.1b and baseline of 0.7b

bsearch7.0ms

end0.0ms

sample17.1s

Algorithm
halfpoints
Results