Average Error: 0.0 → 0.0
Time: 9.6s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019008 +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: 8.9s)Debug log

sample241.0ms

Algorithm
halfpoints

simplify161.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

161.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune24.0ms

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)))

rewrite8.0ms

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)))

series131.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

79.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
51.0ms
(/ (cos y) (exp x))

simplify402.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

147.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
86.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
34.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune582.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.5b
(log1p (fma (cos y) (exp x) (/ (cos y) (exp x))))
0.1b
(expm1 (log1p (fma (cos y) (exp x) (/ (cos y) (exp x)))))
0.0b
(/ (cos y) (exp x))
0.0b
(fma (cos y) (exp x) (/ (cos y) (exp x)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

3.0ms
(/ (cos y) (exp x))
0.0ms
(log1p (fma (cos y) (exp x) (/ (cos y) (exp x))))
0.0ms
(expm1 (log1p (fma (cos y) (exp x) (/ (cos y) (exp x)))))

series457.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

169.0ms
(expm1 (log1p (fma (cos y) (exp x) (/ (cos y) (exp x)))))
132.0ms
(log1p (fma (cos y) (exp x) (/ (cos y) (exp x))))
106.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
49.0ms
(/ (cos y) (exp x))

simplify603.0ms

Counts
39 → 73
Calls

39 calls. Slowest were:

172.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
130.0ms
(- (+ (log 3) (* 1/3 (pow x 2))) (* 1/3 (pow y 2)))
80.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))

prune882.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes66.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample5.3s

Algorithm
halfpoints