Average Error: 0.0 → 0.0
Time: 13.8s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

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 2019002 +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: 13.1s)Debug log

sample170.0ms

Algorithm
halfpoints

simplify137.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

137.0ms
(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

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)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

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

series125.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

82.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
43.0ms
(/ (cos y) (exp x))

simplify291.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

109.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
67.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
33.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune539.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize9.0ms

Local error

Found 3 expressions with local error:

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

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

7.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
4.0ms
(* (cos y) (exp x))
3.0ms
(/ (cos y) (exp x))

series149.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

86.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
32.0ms
(/ (cos y) (exp x))
31.0ms
(* (cos y) (exp x))

simplify571.0ms

Counts
46 → 72
Calls

46 calls. Slowest were:

172.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
111.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
73.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune680.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))
0.0b
(cbrt (exp x))
0.0b
(* (cbrt (exp x)) (cbrt (exp x)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 69
Calls

4 calls. Slowest were:

7.0ms
(* (cbrt (exp x)) (cbrt (exp x)))
1.0ms
(cbrt (exp x))
1.0ms
(cbrt (exp x))

series504.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

152.0ms
(* (cbrt (exp x)) (cbrt (exp x)))
127.0ms
(cbrt (exp x))
114.0ms
(cbrt (exp x))
112.0ms
(cbrt (exp x))

simplify274.0ms

Counts
41 → 81
Calls

41 calls. Slowest were:

60.0ms
(+ (* 1/3 x) (+ (* 1/18 (pow x 2)) 1))
53.0ms
(+ (* 2/3 x) (+ (* 2/9 (pow x 2)) 1))
52.0ms
(+ (* 1/3 x) (+ (* 1/18 (pow x 2)) 1))

prune909.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.0b
(cbrt (exp x))
0.0b
(pow (exp x) 2/3)
0.0b
(/ (cos y) (exp x))
0.0b
(* (* (cos y) (pow (exp x) 2/3)) (cbrt (exp x)))

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 81
Calls

4 calls. Slowest were:

11.0ms
(* (* (cos y) (pow (exp x) 2/3)) (cbrt (exp x)))
3.0ms
(/ (cos y) (exp x))
1.0ms
(pow (exp x) 2/3)

series333.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

123.0ms
(pow (exp x) 2/3)
117.0ms
(cbrt (exp x))
52.0ms
(/ (cos y) (exp x))
41.0ms
(* (* (cos y) (pow (exp x) 2/3)) (cbrt (exp x)))

simplify1.4s

Counts
59 → 93
Calls

59 calls. Slowest were:

399.0ms
(* (* (* (* (cos y) (pow (exp x) 2/3)) (* (cos y) (pow (exp x) 2/3))) (* (cos y) (pow (exp x) 2/3))) (exp x))
330.0ms
(* (* (* (* (cos y) (cos y)) (cos y)) (* (* (pow (exp x) 2/3) (pow (exp x) 2/3)) (pow (exp x) 2/3))) (exp x))
124.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune1.1s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes46.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample5.7s

Algorithm
halfpoints