Average Error: 0.5 → 0.5
Time: 12.5s
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
Target0.0
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 2018365 
(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: 7.2s)Debug log

start89.0ms

Algorithm
intervals

setup17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize34.0ms

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))

rewrite11.0ms

Algorithm
rewrite-expression-head
Counts
3 → 41
Calls

3 calls. Slowest were:

8.0ms
(- (log (+ 1 (exp x))) (* x y))
2.0ms
(log (+ 1 (exp x)))
1.0ms
(+ 1 (exp x))

series99.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

47.0ms
(- (log (+ 1 (exp x))) (* x y))
46.0ms
(log (+ 1 (exp x)))
7.0ms
(+ 1 (exp x))

simplify243.0ms

Counts
21 → 50
Calls

21 calls. Slowest were:

55.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
53.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
37.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

prune524.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.2b

localize16.0ms

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))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
3 → 41
Calls

3 calls. Slowest were:

11.0ms
(- (log (+ (exp x) 1)) (* x y))
5.0ms
(+ (exp x) 1)
4.0ms
(log (+ (exp x) 1))

series151.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

80.0ms
(log (+ (exp x) 1))
60.0ms
(- (log (+ (exp x) 1)) (* x y))
11.0ms
(+ (exp x) 1)

simplify328.0ms

Counts
21 → 50
Calls

21 calls. Slowest were:

59.0ms
(log (- (* (exp x) (exp x)) (* 1 1)))
58.0ms
(log (- (* (exp x) (exp x)) (* 1 1)))
51.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

prune524.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.2b

localize32.0ms

Local error

Found 3 expressions with local error:

7.0b
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
0.0b
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
0.0b
(+ x (+ (* 1/2 (pow x 2)) 2))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
3 → 42
Calls

3 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)))

series88.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

54.0ms
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
25.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
9.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))

simplify2.1s

Counts
21 → 51
Calls

21 calls. Slowest were:

361.0ms
(- (+ (log 1/2) (+ (* 2 (/ 1 (pow x 2))) (* 2 (/ 1 x)))) (* 2 (log (/ 1 x))))
266.0ms
(- (+ (log 1/2) (* 2 (/ 1 x))) (+ (* 2 (log (/ 1 x))) (* x y)))
250.0ms
(- (+ (log 1/2) (+ (* 2 (/ 1 (pow x 2))) (* 2 (/ 1 x)))) (* 2 (log (/ -1 x))))

prune712.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.2b

localize31.0ms

Local error

Found 3 expressions with local error:

0.0b
(- (+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2)))) (* x y))
0.0b
(+ (* 1/2 x) (* 1/8 (pow x 2)))
0.0b
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

rewrite91.0ms

Algorithm
rewrite-expression-head
Counts
3 → 40
Calls

3 calls. Slowest were:

74.0ms
(- (+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2)))) (* x y))
11.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
5.0ms
(+ (* 1/2 x) (* 1/8 (pow x 2)))

series92.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

39.0ms
(- (+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2)))) (* x y))
31.0ms
(+ (* 1/2 x) (* 1/8 (pow x 2)))
23.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

simplify1.1s

Counts
18 → 49
Calls

18 calls. Slowest were:

162.0ms
(/ (exp (+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))) (exp (* x y)))
154.0ms
(/ (* 2 (exp (+ (* 1/2 x) (* 1/8 (pow x 2))))) (exp (* x y)))
132.0ms
(/ (* 2 (* (exp (* 1/2 x)) (exp (* 1/8 (pow x 2))))) (exp (* x y)))

prune795.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.2b

regimes108.0ms

Accuracy

0% (0.2b remaining)

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

bsearch1.0ms