Average Error: 0.5 → 0.5
Time: 13.9s
Precision: 64
Internal Precision: 128
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(e^{x} + 1\right) - y \cdot x\]

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.5
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Taylor expanded around -inf 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019007 
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))

Details

Time bar (total: 13.3s)Debug log

sample139.0ms

Algorithm
intervals

simplify12.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

12.0ms
(- (log (+ 1 (exp x))) (* x y))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize23.0ms

Local error

Found 3 expressions with local error:

0.5b
(log (+ 1 (exp x)))
0.0b
(+ 1 (exp x))
0.0b
(- (log (+ 1 (exp x))) (* x y))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
3 → 41
Calls

3 calls. Slowest were:

4.0ms
(- (log (+ 1 (exp x))) (* x y))
1.0ms
(log (+ 1 (exp x)))
1.0ms
(+ 1 (exp x))

series133.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

84.0ms
(log (+ 1 (exp x)))
43.0ms
(- (log (+ 1 (exp x))) (* x y))
7.0ms
(+ 1 (exp x))

simplify199.0ms

Counts
21 → 50
Calls

21 calls. Slowest were:

45.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
37.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
36.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

prune497.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.4b

localize17.0ms

Local error

Found 3 expressions with local error:

0.5b
(log (+ (exp x) 1))
0.0b
(+ (exp x) 1)
0.0b
(- (log (+ (exp x) 1)) (* x y))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 41
Calls

3 calls. Slowest were:

5.0ms
(- (log (+ (exp x) 1)) (* x y))
2.0ms
(log (+ (exp x) 1))
2.0ms
(+ (exp x) 1)

series109.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

53.0ms
(- (log (+ (exp x) 1)) (* x y))
48.0ms
(log (+ (exp x) 1))
7.0ms
(+ (exp x) 1)

simplify232.0ms

Counts
21 → 50
Calls

21 calls. Slowest were:

43.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
30.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
30.0ms
(log (- (* (exp x) (exp x)) (* 1 1)))

prune405.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.4b

localize24.0ms

Local error

Found 4 expressions with local error:

7.5b
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
0.0b
(+ x (+ (* 1/2 (pow x 2)) 2))
0.0b
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
0.0b
(* 1/2 (pow x 2))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

5.0ms
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
2.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
2.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))

series103.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

48.0ms
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
27.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
19.0ms
(* 1/2 (pow x 2))
9.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

simplify1.6s

Counts
27 → 66
Calls

27 calls. Slowest were:

293.0ms
(- (+ (log 1/2) (+ (* 2 (/ 1 (pow x 2))) (* 2 (/ 1 x)))) (* 2 (log (/ 1 x))))
172.0ms
(- (+ (log 1/2) (* 2 (/ 1 x))) (+ (* 2 (log (/ -1 x))) (* x y)))
162.0ms
(- (+ (log 1/2) (* 2 (/ 1 x))) (+ (* 2 (log (/ 1 x))) (* x y)))

prune674.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.4b

localize13.0ms

Local error

Found 4 expressions with local error:

7.5b
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
7.5b
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
7.5b
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
1.3b
(cbrt (log (+ x (+ (* 1/2 (pow x 2)) 2))))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

2.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
2.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
2.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))

series176.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

76.0ms
(cbrt (log (+ x (+ (* 1/2 (pow x 2)) 2))))
41.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
33.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
25.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))

simplify3.6s

Counts
34 → 66
Calls

34 calls. Slowest were:

401.0ms
(- (+ (* 2/3 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log 1/2) (* 2 (log (/ -1 x)))) 2)) 1/3))) (+ (pow (- (log 1/2) (* 2 (log (/ -1 x)))) 1/3) (* 2/3 (* (/ 1 x) (pow (/ 1 (pow (- (log 1/2) (* 2 (log (/ -1 x)))) 2)) 1/3))))) (* 4/9 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log 1/2) (* 2 (log (/ -1 x)))) 5)) 1/3))))
336.0ms
(- (+ (pow (- (log 1/2) (* 2 (log (/ 1 x)))) 1/3) (+ (* 2/3 (* (/ 1 x) (pow (/ 1 (pow (- (log 1/2) (* 2 (log (/ 1 x)))) 2)) 1/3))) (* 2/3 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log 1/2) (* 2 (log (/ 1 x)))) 2)) 1/3))))) (* 4/9 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log 1/2) (* 2 (log (/ 1 x)))) 5)) 1/3))))
305.0ms
(- (+ (log 1/2) (+ (* 2 (/ 1 (pow x 2))) (* 2 (/ 1 x)))) (* 2 (log (/ 1 x))))

prune761.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.4b

regimes119.0ms

Accuracy

0% (0.2b remaining)

Error of 0.5b against oracle of 0.3b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample4.4s

Algorithm
intervals