Average Error: 39.7 → 0.0
Time: 14.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
↓
\[\sqrt{e^{x} + 1}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.7

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

  2. Initial simplification0.0

    \[\leadsto \sqrt{e^{x} + 1}\]

  3. Final simplification0.0

    \[\leadsto \sqrt{e^{x} + 1}\]

Runtime

Time bar (total: 14.8s)Debug logProfile

start478.0ms

Algorithm
intervals

setup847.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize24.0ms

Local error

Found 1 expressions with local error:

0.0b
(+ (exp x) 1)

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
1 → 15
Calls

1 calls. Slowest were:

5.0ms
(+ (exp x) 1)

series20.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

20.0ms
(+ (exp x) 1)

simplify170.0ms

Counts
5 → 18
Calls

5 calls. Slowest were:

158.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
4.0ms
(+ (exp x) 1)
3.0ms
(+ (exp x) 1)

prune256.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 +o rules:numerics
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  (sqrt (/ (- (exp (* 2 x)) 1) (- (exp x) 1))))