Average Error: 0.0 → 0.0
Time: 17.2s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}\]

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

    \[\leadsto \color{blue}{\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}}\]
  3. Using strategy rm

  4. Applied fma-udef0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(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: 16.7s)Debug log

sample224.0ms

Algorithm
halfpoints
Results

simplify184.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
184.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune18.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 2 expressions with local error:

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

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×times-frac
4×add-cbrt-cube
3×associate-/l*
3×add-exp-log
3×associate-/r*
2×add-log-exp
2×log1p-expm1-u
2×pow1
2×expm1-log1p-u
1×div-inv
1×fma-udef
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
2 → 39
Calls
2 calls:
Slowest
7.0ms
(/ (cos y) (exp x))
0.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

series148.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
95.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
53.0ms
(/ (cos y) (exp x))

simplify448.0ms

Counts
31 → 45
Calls
31 calls:
Slowest
170.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
113.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
31.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))
27.0ms
(- (+ (pow x 2) 2) (pow y 2))
19.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) 1)

prune485.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize8.0ms

Local error

Found 2 expressions with local error:

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

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
9×times-frac
4×add-log-exp
4×add-cbrt-cube
3×associate-/l*
3×add-exp-log
3×associate-/r*
2×div-inv
2×log1p-expm1-u
2×pow1
2×expm1-log1p-u
1×distribute-lft-out
1×flip-+
1×div-exp
1×flip3-+
1×frac-2neg
1×fma-def
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
2 → 44
Calls
2 calls:
Slowest
7.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
3.0ms
(/ (cos y) (exp x))

series148.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
101.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
47.0ms
(/ (cos y) (exp x))

simplify490.0ms

Counts
35 → 50
Calls
35 calls:
Slowest
165.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
115.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
43.0ms
(- (+ (pow x 2) 2) (pow y 2))
31.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))
21.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) 1)

prune518.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize28.0ms

Local error

Found 4 expressions with local error:

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

rewrite24.0ms

Algorithm
rewrite-expression-head
Rules
276×times-frac
160×add-sqr-sqrt
158×add-cube-cbrt
158×*-un-lft-identity
150×sqrt-prod
25×associate-/l*
12×associate-/r*
11×add-exp-log
11×add-cbrt-cube
9×div-inv
4×add-log-exp
4×log1p-expm1-u
4×div-exp
4×cbrt-undiv
4×pow1
4×expm1-log1p-u
2×pow1/2
2×frac-2neg
2×clear-num
2×rem-sqrt-square
1×associate-/l/
Counts
4 → 246
Calls
4 calls:
Slowest
10.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
7.0ms
(/ (cos y) (sqrt (exp x)))
1.0ms
(sqrt (exp x))
1.0ms
(sqrt (exp x))

series151.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
56.0ms
(/ (/ (cos y) (sqrt (exp x))) (sqrt (exp x)))
56.0ms
(/ (cos y) (sqrt (exp x)))
19.0ms
(sqrt (exp x))
19.0ms
(sqrt (exp x))

simplify5.9s

Counts
362 → 258
Calls
362 calls:
Slowest
473.0ms
(/ (* (* (/ (cos y) (sqrt (exp x))) (/ (cos y) (sqrt (exp x)))) (/ (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
221.0ms
(/ (/ (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
152.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) 1)
136.0ms
(- 1 (+ (* 1/2 (pow y 2)) (* 1/2 x)))
134.0ms
(/ (/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (sqrt (exp x))) (cbrt (sqrt (exp x))))) (sqrt 1))

prune2.8s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes68.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample5.0s

Algorithm
halfpoints
Results