Average Error: 0.0 → 0.0
Time: 6.7s
Precision: 64
Internal Precision: 128
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[e^{-1} \cdot {\left(e^{x}\right)}^{x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[e^{-\left(1 - x \cdot x\right)}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{e^{(x \cdot x + -1)_*}}\]
  3. Using strategy rm

  4. Applied fma-udef0.0

    \[\leadsto e^{\color{blue}{x \cdot x + -1}}\]

  5. Applied exp-sum0.0

    \[\leadsto \color{blue}{e^{x \cdot x} \cdot e^{-1}}\]
  6. Using strategy rm

  7. Applied exp-prod0.0

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot e^{-1}\]

  8. Final simplification0.0

    \[\leadsto e^{-1} \cdot {\left(e^{x}\right)}^{x}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))

Details

Time bar (total: 6.4s)Debug log

sample41.0ms

Algorithm
intervals

simplify89.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

89.0ms
(exp (- (- 1 (* x x))))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.1b

localize7.0ms

Local error

Found 1 expressions with local error:

0.1b
(exp (fma x x -1))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

2.0ms
(exp (fma x x -1))

series23.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

23.0ms
(exp (fma x x -1))

simplify225.0ms

Counts
8 → 17
Calls

8 calls. Slowest were:

176.0ms
(+ (exp -1) (+ (* (pow x 2) (exp -1)) (* 1/2 (* (pow x 4) (exp -1)))))
25.0ms
(exp (* (cbrt (fma x x -1)) (cbrt (fma x x -1))))
8.0ms
(exp (sqrt (fma x x -1)))

prune150.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 2 expressions with local error:

0.1b
(exp (* x x))
0.0b
(* (exp (* x x)) (exp -1))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 30
Calls

2 calls. Slowest were:

5.0ms
(* (exp (* x x)) (exp -1))
1.0ms
(exp (* x x))

series149.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

133.0ms
(* (exp (* x x)) (exp -1))
16.0ms
(exp (* x x))

simplify476.0ms

Counts
16 → 36
Calls

16 calls. Slowest were:

191.0ms
(* (* (* (exp (* x x)) (exp (* x x))) (exp (* x x))) (* (* (exp -1) (exp -1)) (exp -1)))
146.0ms
(+ (exp -1) (+ (* (pow x 2) (exp -1)) (* 1/2 (* (pow x 4) (exp -1)))))
50.0ms
(* (exp (* x x)) (exp -1))

prune297.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 2 expressions with local error:

0.0b
(* (pow (exp x) x) (exp -1))
0.0b
(pow (exp x) x)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 38
Calls

2 calls. Slowest were:

5.0ms
(* (pow (exp x) x) (exp -1))
1.0ms
(pow (exp x) x)

series200.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

133.0ms
(* (pow (exp x) x) (exp -1))
67.0ms
(pow (exp x) x)

simplify449.0ms

Counts
23 → 44
Calls

23 calls. Slowest were:

184.0ms
(+ (exp -1) (+ (* (pow x 2) (exp -1)) (* 1/2 (* (pow x 4) (exp -1)))))
71.0ms
(* (* (* (pow (exp x) x) (pow (exp x) x)) (pow (exp x) x)) (* (* (exp -1) (exp -1)) (exp -1)))
67.0ms
(+ (* (log (exp x)) x) -1)

prune415.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 3 expressions with local error:

0.0b
(pow (exp x) x)
0.0b
(* (* (pow (exp x) x) (sqrt (exp -1))) (sqrt (exp -1)))
0.0b
(* (pow (exp x) x) (sqrt (exp -1)))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
3 → 65
Calls

3 calls. Slowest were:

17.0ms
(* (* (pow (exp x) x) (sqrt (exp -1))) (sqrt (exp -1)))
11.0ms
(* (pow (exp x) x) (sqrt (exp -1)))
1.0ms
(pow (exp x) x)

series366.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

162.0ms
(* (pow (exp x) x) (sqrt (exp -1)))
134.0ms
(* (* (pow (exp x) x) (sqrt (exp -1))) (sqrt (exp -1)))
69.0ms
(pow (exp x) x)

simplify1.5s

Counts
43 → 74
Calls

43 calls. Slowest were:

396.0ms
(* (* (* (* (pow (exp x) x) (sqrt (exp -1))) (* (pow (exp x) x) (sqrt (exp -1)))) (* (pow (exp x) x) (sqrt (exp -1)))) (* (* (sqrt (exp -1)) (sqrt (exp -1))) (sqrt (exp -1))))
357.0ms
(* (* (* (* (pow (exp x) x) (pow (exp x) x)) (pow (exp x) x)) (* (* (sqrt (exp -1)) (sqrt (exp -1))) (sqrt (exp -1)))) (* (* (sqrt (exp -1)) (sqrt (exp -1))) (sqrt (exp -1))))
293.0ms
(+ (* 1/2 (* (pow x 4) (sqrt (exp -1)))) (+ (* (pow x 2) (sqrt (exp -1))) (sqrt (exp -1))))

prune670.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes29.0ms

Accuracy

0% (0.0b remaining)

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

bsearch3.0ms

end0.0ms

sample1.2s

Algorithm
intervals