Average Error: 0.0 → 0.0
Time: 36.1s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

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. Final simplification0.0

    \[\leadsto \Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019004 
(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: 35.0s)Debug log

sample189.0ms

Algorithm
halfpoints

simplify140.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

140.0ms
(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

localize49.0ms

Local error

Found 4 expressions with local error:

16.2b
(- (exp x) (exp (- x)))
0.0b
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
0.0b
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
0.0b
(+ (exp x) (exp (- x)))

rewrite28.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

10.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
10.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
4.0ms
(- (exp x) (exp (- x)))

series231.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
71.0ms
(* (/ (+ (exp x) (exp (- x))) 2) (cos y))
44.0ms
(- (exp x) (exp (- x)))
41.0ms
(+ (exp x) (exp (- x)))

simplify2.1s

Counts
38 → 76
Calls

38 calls. Slowest were:

495.0ms
(* (* (* (/ (- (exp x) (exp (- x))) 2) (/ (- (exp x) (exp (- x))) 2)) (/ (- (exp x) (exp (- x))) 2)) (* (* (sin y) (sin y)) (sin y)))
266.0ms
(* 1/2 (* (- (exp x) (exp (* -1 x))) (sin y)))
257.0ms
(* (* (* (/ (+ (exp x) (exp (- x))) 2) (/ (+ (exp x) (exp (- x))) 2)) (/ (+ (exp x) (exp (- x))) 2)) (* (* (cos y) (cos y)) (cos y)))

prune848.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

16.2b
(- (exp x) (exp (- x)))
0.5b
(cbrt (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
0.0b
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
0.0b
(* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x))))

rewrite71.0ms

Algorithm
rewrite-expression-head
Counts
4 → 123
Calls

4 calls. Slowest were:

32.0ms
(* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x))))
23.0ms
(cbrt (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
10.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))

series170.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(* (/ (- (exp x) (exp (- x))) 2) (sin y))
38.0ms
(- (exp x) (exp (- x)))
36.0ms
(* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x))))
24.0ms
(cbrt (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))

simplify21.5s

Counts
119 → 135
Calls

119 calls. Slowest were:

950.0ms
(* (* (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (+ (pow (exp x) 3) (pow (exp (- x)) 3)))
689.0ms
(* (* (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x))))) (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x))))) (* (* (+ (exp x) (exp (- x))) (+ (exp x) (exp (- x)))) (+ (exp x) (exp (- x)))))
667.0ms
(* (* (+ (pow (exp x) 3) (pow (exp (- x)) 3)) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x))))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp (- x)))))

prune2.4s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes106.0ms

Accuracy

0% (0.0b remaining)

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

bsearch2.0ms

end0.0ms

sample7.2s

Algorithm
halfpoints