Average Error: 0.0 → 0.0
Time: 10.7s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{(\left(e^{x}\right) \cdot \left(\cos y\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}\]

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. Initial simplification0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2018360 +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: 4.9s)Debug log

start189.0ms

Algorithm
halfpoints

setup157.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize35.0ms

Local error

Found 2 expressions with local error:

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

series129.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

68.0ms
(fma (exp x) (cos y) (/ (cos y) (exp x)))
60.0ms
(/ (cos y) (exp x))

simplify420.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

124.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
118.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
41.0ms
(- (+ (pow x 2) 2) (pow y 2))

prune540.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize30.0ms

Local error

Found 4 expressions with local error:

0.3b
(cbrt (/ (cos y) (exp x)))
0.3b
(cbrt (/ (cos y) (exp x)))
0.3b
(cbrt (/ (cos y) (exp x)))
0.3b
(* (cbrt (/ (cos y) (exp x))) (cbrt (/ (cos y) (exp x))))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

12.0ms
(* (cbrt (/ (cos y) (exp x))) (cbrt (/ (cos y) (exp x))))
2.0ms
(cbrt (/ (cos y) (exp x)))
2.0ms
(cbrt (/ (cos y) (exp x)))

series404.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

122.0ms
(cbrt (/ (cos y) (exp x)))
115.0ms
(cbrt (/ (cos y) (exp x)))
84.0ms
(* (cbrt (/ (cos y) (exp x))) (cbrt (/ (cos y) (exp x))))
83.0ms
(cbrt (/ (cos y) (exp x)))

simplify831.0ms

Counts
98 → 137
Calls

98 calls. Slowest were:

145.0ms
(- 1 (+ (* 1/6 (pow y 2)) (* 1/3 x)))
131.0ms
(- 1 (+ (* 1/6 (pow y 2)) (* 1/3 x)))
101.0ms
(- 1 (+ (* 1/6 (pow y 2)) (* 1/3 x)))

prune2.1s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize11.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune0.0ms

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes79.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms