Average Error: 0.5 → 0.4
Time: 7.5s
Precision: 64
Internal Precision: 128
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log_* (1 + e^{x}) - y \cdot x\]

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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.4

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

  3. Final simplification0.4

    \[\leadsto \log_* (1 + e^{x}) - y \cdot x\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(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: 7.1s)Debug log

sample140.0ms

Algorithm
intervals

simplify17.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

17.0ms
(- (log (+ 1 (exp x))) (* x y))

prune34.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 2 expressions with local error:

0.0b
(- (log1p (exp x)) (* y x))
0.0b
(log1p (exp x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 32
Calls

2 calls. Slowest were:

5.0ms
(- (log1p (exp x)) (* y x))
0.0ms
(log1p (exp x))

series111.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

73.0ms
(log1p (exp x))
37.0ms
(- (log1p (exp x)) (* y x))

simplify264.0ms

Counts
17 → 38
Calls

17 calls. Slowest were:

67.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
34.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
26.0ms
(/ (exp (log1p (exp x))) (exp (* y x)))

prune362.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

1.4b
(log (exp (- (log1p (exp x)) (* y x))))
0.0b
(exp (- (log1p (exp x)) (* y x)))
0.0b
(- (log1p (exp x)) (* y x))
0.0b
(log1p (exp x))

rewrite24.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

12.0ms
(log (exp (- (log1p (exp x)) (* y x))))
7.0ms
(exp (- (log1p (exp x)) (* y x)))
3.0ms
(- (log1p (exp x)) (* y x))

series218.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(log1p (exp x))
54.0ms
(log (exp (- (log1p (exp x)) (* y x))))
48.0ms
(- (log1p (exp x)) (* y x))
31.0ms
(exp (- (log1p (exp x)) (* y x)))

simplify816.0ms

Counts
48 → 88
Calls

48 calls. Slowest were:

63.0ms
(log (exp (fma (sqrt (log1p (exp x))) (sqrt (log1p (exp x))) (- (* x y)))))
60.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
56.0ms
(exp (fma (* (cbrt (log1p (exp x))) (cbrt (log1p (exp x)))) (cbrt (log1p (exp x))) (- (* x y))))

prune858.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0.0b

regimes54.0ms

Accuracy

0% (0.0b remaining)

Error of 0.4b against oracle of 0.4b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample4.1s

Algorithm
intervals