Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\cos y \cdot (e^{\log_* (1 + \frac{1}{e^{x}})} - 1)^*\right))_*}{2}\]

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. Initial simplification0.0

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

  4. Applied div-inv0.0

    \[\leadsto \frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \color{blue}{\left(\cos y \cdot \frac{1}{e^{x}}\right)})_*}{2}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.0

    \[\leadsto \frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\cos y \cdot \color{blue}{(e^{\log_* (1 + \frac{1}{e^{x}})} - 1)^*}\right))_*}{2}\]

  7. Final simplification0.0

    \[\leadsto \frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\cos y \cdot (e^{\log_* (1 + \frac{1}{e^{x}})} - 1)^*\right))_*}{2}\]

Reproduce

herbie shell --seed 2018362 +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: 4.5s)Debug log

start206.0ms

Algorithm
halfpoints

setup153.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize32.0ms

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
Counts
2 → 39
Calls

2 calls. Slowest were:

7.0ms
(/ (cos y) (exp x))
0.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

series150.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

97.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
52.0ms
(/ (cos y) (exp x))

simplify273.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

120.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
61.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
20.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune581.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0b

localize26.0ms

Local error

Found 3 expressions with local error:

0.0b
(/ 1 (exp x))
0.0b
(fma (cos y) (exp x) (* (cos y) (/ 1 (exp x))))
0.0b
(* (cos y) (/ 1 (exp x)))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 51
Calls

3 calls. Slowest were:

8.0ms
(* (cos y) (/ 1 (exp x)))
2.0ms
(/ 1 (exp x))
0.0ms
(fma (cos y) (exp x) (* (cos y) (/ 1 (exp x))))

series118.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

64.0ms
(fma (cos y) (exp x) (* (cos y) (/ 1 (exp x))))
43.0ms
(* (cos y) (/ 1 (exp x)))
10.0ms
(/ 1 (exp x))

simplify742.0ms

Counts
25 → 60
Calls

25 calls. Slowest were:

489.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (/ 1 (exp x)) (/ 1 (exp x))) (/ 1 (exp x))))
101.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
38.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune699.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize34.0ms

Local error

Found 4 expressions with local error:

0.0b
(/ 1 (exp x))
0.0b
(fma (cos y) (exp x) (* (cos y) (expm1 (log1p (/ 1 (exp x))))))
0.0b
(expm1 (log1p (/ 1 (exp x))))
0.0b
(log1p (/ 1 (exp x)))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

2.0ms
(/ 1 (exp x))
1.0ms
(log1p (/ 1 (exp x)))
0.0ms
(expm1 (log1p (/ 1 (exp x))))

series344.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

138.0ms
(fma (cos y) (exp x) (* (cos y) (expm1 (log1p (/ 1 (exp x))))))
128.0ms
(expm1 (log1p (/ 1 (exp x))))
65.0ms
(log1p (/ 1 (exp x)))
14.0ms
(/ 1 (exp x))

simplify240.0ms

Counts
17 → 62
Calls

17 calls. Slowest were:

67.0ms
(- (+ (log 2) (* 1/8 (pow x 2))) (* 1/2 x))
60.0ms
(- (+ (* 1/2 (pow x 2)) 1) x)
51.0ms
(- (+ (* 1/2 (pow x 2)) 1) x)

prune868.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b