Average Error: 0.0 → 0.0
Time: 20.4s
Precision: 64
Internal Precision: 576
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\cos y \cdot e^{-x} + e^{x} \cdot \cos y}{2}\]

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

    \[\leadsto \color{blue}{\frac{\frac{\cos y}{e^{x}} + \cos y \cdot e^{x}}{2}}\]
  3. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot \frac{1}{e^{x}}} + \cos y \cdot e^{x}}{2}\]

  5. Applied distribute-lft-out0.0

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

  6. Simplified0.0

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

  8. Applied distribute-lft-in0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot e^{-x} + \cos y \cdot e^{x}}}{2}\]

  9. Final simplification0.0

    \[\leadsto \frac{\cos y \cdot e^{-x} + e^{x} \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 1042149663 
(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: 19.2s)Debug log

sample378.0ms

Algorithm
halfpoints
Results

simplify255.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
255.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune36.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize62.0ms

Local error

Found 3 expressions with local error:

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

rewrite54.0ms

Algorithm
rewrite-expression-head
Rules
13×add-cube-cbrt
13×*-un-lft-identity
13×add-sqr-sqrt
9×times-frac
7×add-cbrt-cube
5×add-log-exp
5×add-exp-log
5×pow1
3×associate-/l*
3×associate-*l*
3×associate-*r*
3×associate-/r*
2×div-inv
1×distribute-lft-out
1×flip-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×div-exp
1×flip3-+
1×frac-2neg
1×sum-log
1×clear-num
1×+-commutative
1×cbrt-undiv
Counts
3 → 56
Calls
3 calls:
Slowest
22.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
15.0ms
(/ (cos y) (exp x))
15.0ms
(* (cos y) (exp x))

series402.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
211.0ms
(+ (/ (cos y) (exp x)) (* (cos y) (exp x)))
101.0ms
(/ (cos y) (exp x))
89.0ms
(* (cos y) (exp x))

simplify1.1s

Counts
45 → 65
Calls
45 calls:
Slowest
278.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
168.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
158.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
105.0ms
(- (+ x 1) (* 1/2 (pow y 2)))
70.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune1.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize35.0ms

Local error

Found 2 expressions with local error:

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

rewrite45.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×associate-*r*
4×add-cbrt-cube
4×pow1
4×add-sqr-sqrt
3×associate-*l*
2×distribute-lft-out
2×flip-+
2×associate-*r/
2×flip3-+
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×distribute-lft-in
1×sum-log
1×+-commutative
1×distribute-rgt-in
Counts
2 → 34
Calls
2 calls:
Slowest
27.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
17.0ms
(+ (exp (- x)) (exp x))

series159.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
107.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
51.0ms
(+ (exp (- x)) (exp x))

simplify1.0s

Counts
21 → 40
Calls
21 calls:
Slowest
290.0ms
(+ (* (exp (- x)) (exp (- x))) (- (* (exp x) (exp x)) (* (exp (- x)) (exp x))))
252.0ms
(* (cos y) (+ (exp (- x)) (exp x)))
164.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (+ (exp (- x)) (exp x)) (+ (exp (- x)) (exp x))) (+ (exp (- x)) (exp x))))
74.0ms
(* (+ (exp (* -1 x)) (exp x)) (cos y))
61.0ms
(* (+ (exp x) (exp (- x))) (cos y))

prune721.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize22.0ms

Local error

Found 3 expressions with local error:

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

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×add-cbrt-cube
7×*-un-lft-identity
7×pow1
7×add-sqr-sqrt
6×associate-*l*
6×associate-*r*
5×add-log-exp
5×add-exp-log
2×cbrt-unprod
2×*-commutative
2×associate-*r/
2×prod-exp
2×pow-prod-down
2×exp-neg
1×distribute-lft-out
1×flip-+
1×un-div-inv
1×exp-diff
1×flip3-+
1×sum-log
1×+-commutative
1×neg-sub0
Counts
3 → 49
Calls
3 calls:
Slowest
18.0ms
(+ (* (cos y) (exp (- x))) (* (cos y) (exp x)))
14.0ms
(* (cos y) (exp (- x)))
9.0ms
(* (cos y) (exp x))

series487.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
283.0ms
(+ (* (cos y) (exp (- x))) (* (cos y) (exp x)))
116.0ms
(* (cos y) (exp x))
88.0ms
(* (cos y) (exp (- x)))

simplify2.1s

Counts
31 → 58
Calls
31 calls:
Slowest
640.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp (- x)) (exp (- x))) (exp (- x))))
363.0ms
(* (exp (* (cos y) (exp (- x)))) (exp (* (cos y) (exp x))))
308.0ms
(+ (* (cos y) (exp (- x))) (* (exp x) (cos y)))
156.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
149.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))

prune1.3s

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes136.0ms

Accuracy

0% (0.0b remaining)

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

bsearch4.0ms

end0.0ms

sample9.6s

Algorithm
halfpoints
Results