Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
Internal Precision: 128
\[e^{-\left(1 - x \cdot x\right)}\]
↓
\[\frac{{\left(\sqrt{e^{x}}\right)}^{x}}{e^{\log \left(\frac{e}{{\left(\sqrt{e^{x}}\right)}^{x}}\right)}}\]

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

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

  4. Applied exp-prod0.0

    \[\leadsto \frac{\color{blue}{{\left(e^{x}\right)}^{x}}}{e}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Applied unpow-prod-down0.0

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

  8. Applied associate-/l*0.0

    \[\leadsto \color{blue}{\frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\frac{e}{{\left(\sqrt{e^{x}}\right)}^{x}}}}\]
  9. Using strategy rm

  10. Applied add-exp-log0.0

    \[\leadsto \frac{{\left(\sqrt{e^{x}}\right)}^{x}}{\color{blue}{e^{\log \left(\frac{e}{{\left(\sqrt{e^{x}}\right)}^{x}}\right)}}}\]

  11. Final simplification0.0

    \[\leadsto \frac{{\left(\sqrt{e^{x}}\right)}^{x}}{e^{\log \left(\frac{e}{{\left(\sqrt{e^{x}}\right)}^{x}}\right)}}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (x)
  :name "exp neg sub"
  (exp (- (- 1 (* x x)))))

Details

Time bar (total: 7.1s)Debug log

sample33.0ms

Algorithm
intervals
Results
24.0ms495×body80valid

simplify56.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
56.0ms
(exp (- (- 1 (* x x))))

prune11.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

0.0b
(exp (* x x))
0.0b
(/ (exp (* x x)) E)

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
3×add-log-exp
3×associate-/l*
3×add-cube-cbrt
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-exp-log
2×add-cbrt-cube
2×pow1
1×rem-exp-log
1×div-inv
1×frac-2neg
1×clear-num
1×exp-prod
Counts
2 → 22
Calls
2 calls:
Slowest
5.0ms
(/ (exp (* x x)) E)
1.0ms
(exp (* x x))

series55.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
40.0ms
(/ (exp (* x x)) E)
15.0ms
(exp (* x x))

simplify60.0ms

Counts
10 → 28
Calls
10 calls:
Slowest
31.0ms
(+ (* 1/2 (/ (pow x 4) E)) (+ (/ (pow x 2) E) (/ 1 E)))
9.0ms
(+ (pow x 2) (+ (* 1/2 (pow x 4)) 1))
5.0ms
(/ (exp (pow x 2)) E)
5.0ms
(/ (exp (pow x 2)) E)
2.0ms
(exp (pow x 2))

prune160.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 2 expressions with local error:

0.0b
(/ (pow (exp x) x) E)
0.0b
(pow (exp x) x)

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
6×associate-/l*
6×unpow-prod-down
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
3×pow1
2×add-log-exp
2×add-exp-log
2×add-cbrt-cube
1×div-inv
1×pow-exp
1×frac-2neg
1×pow-pow
1×clear-num
1×pow-to-exp
Counts
2 → 29
Calls
2 calls:
Slowest
5.0ms
(/ (pow (exp x) x) E)
1.0ms
(pow (exp x) x)

series135.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
81.0ms
(/ (pow (exp x) x) E)
54.0ms
(pow (exp x) x)

simplify58.0ms

Counts
16 → 35
Calls
16 calls:
Slowest
26.0ms
(+ (* 1/2 (/ (pow x 4) E)) (+ (/ (pow x 2) E) (/ 1 E)))
10.0ms
(+ (pow x 2) (+ (* 1/2 (pow x 4)) 1))
5.0ms
(/ (exp (pow x 2)) E)
4.0ms
(/ (exp (pow x 2)) E)
2.0ms
(exp (pow x 2))

prune222.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.0b
(pow (sqrt (exp x)) x)
0.0b
(pow (sqrt (exp x)) x)
0.0b
(/ (pow (sqrt (exp x)) x) (/ E (pow (sqrt (exp x)) x)))
0.0b
(/ E (pow (sqrt (exp x)) x))

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
48×unpow-prod-down
36×add-cube-cbrt
36×times-frac
36×*-un-lft-identity
36×add-sqr-sqrt
24×sqrt-prod
13×associate-/r*
12×div-inv
11×add-exp-log
9×associate-/l*
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×pow-pow
3×pow-exp
3×div-exp
3×pow-to-exp
2×pow1/2
2×frac-2neg
2×clear-num
1×associate-/r/
1×cbrt-undiv
Counts
4 → 117
Calls
4 calls:
Slowest
9.0ms
(/ (pow (sqrt (exp x)) x) (/ E (pow (sqrt (exp x)) x)))
2.0ms
(pow (sqrt (exp x)) x)
2.0ms
(pow (sqrt (exp x)) x)
2.0ms
(/ E (pow (sqrt (exp x)) x))

series312.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
107.0ms
(/ (pow (sqrt (exp x)) x) (/ E (pow (sqrt (exp x)) x)))
91.0ms
(/ E (pow (sqrt (exp x)) x))
58.0ms
(pow (sqrt (exp x)) x)
55.0ms
(pow (sqrt (exp x)) x)

simplify1.4s

Counts
128 → 129
Calls
128 calls:
Slowest
313.0ms
(/ (* (* (pow (sqrt (exp x)) x) (pow (sqrt (exp x)) x)) (pow (sqrt (exp x)) x)) (* (* (/ E (pow (sqrt (exp x)) x)) (/ E (pow (sqrt (exp x)) x))) (/ E (pow (sqrt (exp x)) x))))
183.0ms
(- (+ (* 1/8 (* (pow x 4) E)) E) (* 1/2 (* (pow x 2) E)))
60.0ms
(- (* (log (sqrt (exp x))) x) (log (/ E (pow (sqrt (exp x)) x))))
46.0ms
(- (log (pow (sqrt (exp x)) x)) (log (/ E (pow (sqrt (exp x)) x))))
36.0ms
(+ (* 1/2 (/ (pow x 4) E)) (+ (/ (pow x 2) E) (/ 1 E)))

prune1.1s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 4 expressions with local error:

0.0b
(exp (log (/ E (pow (sqrt (exp x)) x))))
0.0b
(pow (sqrt (exp x)) x)
0.0b
(pow (sqrt (exp x)) x)
0.0b
(log (/ E (pow (sqrt (exp x)) x)))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
12×unpow-prod-down
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
8×log-prod
8×pow1
7×add-exp-log
6×sqrt-prod
4×add-log-exp
4×exp-sum
4×add-cbrt-cube
4×pow-pow
4×exp-prod
2×div-inv
2×log-pow
2×pow-exp
2×pow1/2
2×log-div
2×pow-to-exp
1×rem-exp-log
1×rem-log-exp
1×exp-diff
Counts
4 → 65
Calls
4 calls:
Slowest
5.0ms
(exp (log (/ E (pow (sqrt (exp x)) x))))
3.0ms
(pow (sqrt (exp x)) x)
3.0ms
(pow (sqrt (exp x)) x)
2.0ms
(log (/ E (pow (sqrt (exp x)) x)))

series706.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
504.0ms
(log (/ E (pow (sqrt (exp x)) x)))
84.0ms
(exp (log (/ E (pow (sqrt (exp x)) x))))
68.0ms
(pow (sqrt (exp x)) x)
50.0ms
(pow (sqrt (exp x)) x)

simplify425.0ms

Counts
45 → 77
Calls
45 calls:
Slowest
205.0ms
(- (+ (* 1/8 (* (pow x 4) E)) E) (* 1/2 (* (pow x 2) E)))
19.0ms
(+ (* 1/2 (pow x 2)) (+ (* 1/8 (pow x 4)) 1))
17.0ms
(log (/ E (exp (* (log (sqrt (exp x))) x))))
16.0ms
(exp (* (cbrt (log (/ E (pow (sqrt (exp x)) x)))) (cbrt (log (/ E (pow (sqrt (exp x)) x))))))
16.0ms
(log (/ E (exp (* (log (sqrt (exp x))) x))))

prune917.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes49.0ms

Accuracy

0% (0.0b remaining)

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

bsearch1.0ms

end0.0ms

sample1.3s

Algorithm
intervals
Results
1.1s15909×body80valid