Average Error: 58.8 → 0.0
Time: 4.7s
Precision: 64
Internal Precision: 128
\[e^{x} - 1\]
↓
\[(e^{x} - 1)^*\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.4
Herbie0.0
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  2. Simplified0.0

    \[\leadsto \color{blue}{(e^{x} - 1)^*}\]

  3. Final simplification0.0

    \[\leadsto (e^{x} - 1)^*\]

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))

Details

Time bar (total: 4.4s)Debug log

sample111.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
2.0ms
(- (exp x) 1)

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize3.0ms

Local error

Found 1 expressions with local error:

0.0b
(expm1 x)

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×expm1-udef
1×log1p-expm1-u
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(expm1 x)

series12.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
12.0ms
(expm1 x)

simplify165.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
158.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
3.0ms
(- (exp x) 1)
2.0ms
(- (exp x) 1)

prune104.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize5.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (sqrt (expm1 x)) (sqrt (expm1 x)))
0.0b
(expm1 x)
0.0b
(expm1 x)

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
8×pow1
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×sqrt-prod
6×associate-*l*
6×associate-*r*
5×add-exp-log
5×add-cbrt-cube
4×pow1/2
3×add-log-exp
3×log1p-expm1-u
3×expm1-log1p-u
2×expm1-udef
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×pow2
Counts
3 → 52
Calls
3 calls:
Slowest
5.0ms
(* (sqrt (expm1 x)) (sqrt (expm1 x)))
0.0ms
(expm1 x)
0.0ms
(expm1 x)

series41.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
17.0ms
(expm1 x)
12.0ms
(* (sqrt (expm1 x)) (sqrt (expm1 x)))
12.0ms
(expm1 x)

simplify649.0ms

Counts
30 → 61
Calls
30 calls:
Slowest
209.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
202.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
145.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
34.0ms
(* (* (* (sqrt (expm1 x)) (sqrt (expm1 x))) (sqrt (expm1 x))) (* (* (sqrt (expm1 x)) (sqrt (expm1 x))) (sqrt (expm1 x))))
4.0ms
(- (exp x) 1)

prune452.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes33.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample2.8s

Algorithm
intervals