Average Error: 43.6 → 0.7
Time: 31.2s
Precision: 64
Internal Precision: 1344
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{(\left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) \cdot x + \left(\frac{1}{60} \cdot {x}^{5}\right))_*}{2} \cdot \sin y i\right))\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.6

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

  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. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(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: 27.7s)Debug log

sample444.0ms

Algorithm
halfpoints
Results

simplify105.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
105.0ms
(im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 42.7b

localize116.0ms

Local error

Found 3 expressions with local error:

57.0b
(- (exp x) (exp (- x)))
0.4b
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
0.0b
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
16×*-un-lft-identity
16×add-sqr-sqrt
14×add-cube-cbrt
9×prod-diff
8×associate-*l*
7×add-exp-log
7×add-cbrt-cube
7×pow1
6×associate-*r*
5×add-log-exp
3×log1p-expm1-u
3×fma-neg
3×expm1-log1p-u
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×flip--
1×diff-log
1×flip3--
1×sub-neg
Counts
3 → 70
Calls
3 calls:
Slowest
10.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
10.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
5.0ms
(- (exp x) (exp (- x)))

series244.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
126.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
66.0ms
(- (exp x) (exp (- x)))
52.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))

simplify3.0s

Counts
53 → 79
Calls
53 calls:
Slowest
453.0ms
(* (* (* (/ (- (exp x) (exp (- x))) 2) (/ (- (exp x) (exp (- x))) 2)) (/ (- (exp x) (exp (- x))) 2)) (* (* (sin y) (sin y)) (sin y)))
360.0ms
(* 1/2 (* (- (exp x) (exp (- x))) (sin y)))
311.0ms
(* 1/2 (* (- (exp x) (exp (* -1 x))) (sin y)))
271.0ms
(* (* (* (/ (+ (exp x) (exp (- x))) 2) (/ (+ (exp x) (exp (- x))) 2)) (/ (+ (exp x) (exp (- x))) 2)) (* (* (cos y) (cos y)) (cos y)))
260.0ms
(* 1/2 (* (+ (exp (* -1 x)) (exp x)) (cos y)))

prune936.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.4b

localize51.0ms

Local error

Found 4 expressions with local error:

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

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
11×associate-*l*
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
8×add-exp-log
8×add-cbrt-cube
8×pow1
6×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×*-commutative
2×div-inv
2×cbrt-unprod
2×prod-exp
2×associate-*l/
2×pow-prod-down
1×fma-udef
Counts
4 → 65
Calls
4 calls:
Slowest
10.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
5.0ms
(* (/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) 2) (sin y))
2.0ms
(* (pow x 5) 1/60)
0.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))

series233.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
84.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
79.0ms
(* (/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) 2) (sin y))
38.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
32.0ms
(* (pow x 5) 1/60)

simplify2.1s

Counts
34 → 77
Calls
34 calls:
Slowest
308.0ms
(* (* (* (/ (+ (exp x) (exp (- x))) 2) (/ (+ (exp x) (exp (- x))) 2)) (/ (+ (exp x) (exp (- x))) 2)) (* (* (cos y) (cos y)) (cos y)))
239.0ms
(* 1/2 (* (+ (exp (* -1 x)) (exp x)) (cos y)))
224.0ms
(* (* (* (/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) 2) (/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) 2)) (/ (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) 2)) (* (* (sin y) (sin y)) (sin y)))
220.0ms
(* 1/2 (* (+ (exp x) (exp (- x))) (cos y)))
177.0ms
(* (fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60)) (sin y))

prune1.6s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize32.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.4b
(* (cbrt (sin y)) (cbrt (sin y)))

rewrite10.0ms

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×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 69
Calls
4 calls:
Slowest
6.0ms
(* (cbrt (sin y)) (cbrt (sin y)))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))

series641.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
175.0ms
(* (cbrt (sin y)) (cbrt (sin y)))
172.0ms
(cbrt (sin y))
153.0ms
(cbrt (sin y))
140.0ms
(cbrt (sin y))

simplify425.0ms

Counts
41 → 81
Calls
41 calls:
Slowest
109.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
91.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
88.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
66.0ms
(- (+ (pow y 2/3) (* 1/405 (pow (pow y 14) 1/3))) (* 1/9 (pow (pow y 8) 1/3)))
3.0ms
(cbrt (sin y))

prune1.8s

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0.3b

localize3.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.6b
(cbrt (sin y))
0.4b
(* (cbrt (sin y)) (cbrt (sin y)))

rewrite8.0ms

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×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 69
Calls
4 calls:
Slowest
5.0ms
(* (cbrt (sin y)) (cbrt (sin y)))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))
1.0ms
(cbrt (sin y))

series656.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
193.0ms
(cbrt (sin y))
179.0ms
(* (cbrt (sin y)) (cbrt (sin y)))
149.0ms
(cbrt (sin y))
134.0ms
(cbrt (sin y))

simplify450.0ms

Counts
41 → 81
Calls
41 calls:
Slowest
107.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
106.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
84.0ms
(- (pow y 1/3) (+ (* 1/18 (pow (pow y 7) 1/3)) (* 1/3240 (pow (pow y 13) 1/3))))
80.0ms
(- (+ (pow y 2/3) (* 1/405 (pow (pow y 14) 1/3))) (* 1/9 (pow (pow y 8) 1/3)))
4.0ms
(pow (pow (sin y) 2) 1/3)

prune2.0s

Pruning

12 alts after pruning (10 fresh and 2 done)

Merged error: 0.3b

regimes182.0ms

Accuracy

0% (0.7b remaining)

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

bsearch2.0ms

end0.0ms

sample12.7s

Algorithm
halfpoints
Results