Average Error: 0.0 → 0.0
Time: 9.2s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{\cos y}{e^{x}} + \sqrt{e^{x}} \cdot \left(\cos y \cdot \sqrt{e^{x}}\right)}{2}\]

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. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Applied associate-*r*0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019022 +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: 8.7s)Debug log

sample207.0ms

Algorithm
halfpoints
Results

simplify158.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
158.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 1 expressions with local error:

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

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
1.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

series117.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
117.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

simplify27.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
19.0ms
(- (+ (pow x 2) 2) (pow y 2))
4.0ms
(+ (/ (cos y) (exp x)) (* (exp x) (cos y)))
4.0ms
(+ (/ (cos y) (exp x)) (* (exp x) (cos y)))

prune93.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize7.0ms

Local error

Found 2 expressions with local error:

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

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
3×add-exp-log
3×associate-*l*
3×associate-*r*
2×log1p-expm1-u
2×expm1-log1p-u
1×distribute-lft-out
1×div-inv
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×flip3-+
1×fma-def
1×sum-log
1×+-commutative
Counts
2 → 34
Calls
2 calls:
Slowest
11.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
4.0ms
(* (cos y) (exp x))

series149.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
100.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
48.0ms
(* (cos y) (exp x))

simplify223.0ms

Counts
18 → 40
Calls
18 calls:
Slowest
91.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
51.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
33.0ms
(- (+ (pow x 2) 2) (pow y 2))
12.0ms
(* (exp (* (cos y) (exp x))) (exp (/ (cos y) (exp x))))
5.0ms
(+ (/ (cos y) (exp x)) (* (exp x) (cos y)))

prune411.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

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

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
13×add-sqr-sqrt
12×sqrt-prod
12×associate-*r*
11×add-cube-cbrt
11×add-exp-log
11×add-cbrt-cube
11×*-un-lft-identity
11×pow1
4×add-log-exp
4×log1p-expm1-u
4×cbrt-unprod
4×prod-exp
4×pow-prod-down
4×associate-*l*
4×expm1-log1p-u
2×*-commutative
2×pow1/2
2×rem-sqrt-square
Counts
4 → 73
Calls
4 calls:
Slowest
14.0ms
(* (* (cos y) (sqrt (exp x))) (sqrt (exp x)))
8.0ms
(* (cos y) (sqrt (exp x)))
1.0ms
(sqrt (exp x))
1.0ms
(sqrt (exp x))

series131.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
62.0ms
(* (cos y) (sqrt (exp x)))
39.0ms
(* (* (cos y) (sqrt (exp x))) (sqrt (exp x)))
17.0ms
(sqrt (exp x))
12.0ms
(sqrt (exp x))

simplify1.1s

Counts
44 → 85
Calls
44 calls:
Slowest
494.0ms
(* (* (* (* (cos y) (sqrt (exp x))) (* (cos y) (sqrt (exp x)))) (* (cos y) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
253.0ms
(* (* (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x)))) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))
80.0ms
(- (+ (* 1/2 x) 1) (* 1/2 (pow y 2)))
65.0ms
(+ (* 1/2 x) (+ (* 1/8 (pow x 2)) 1))
65.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (sqrt (exp x)) (sqrt (exp x))) (sqrt (exp x))))

prune909.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample5.0s

Algorithm
halfpoints
Results