Average Error: 0.5 → 0.4
Time: 13.8s
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 2019008 +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: 13.1s)Debug log

sample126.0ms

Algorithm
intervals

simplify18.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

18.0ms
(- (log (+ 1 (exp x))) (* x y))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize22.0ms

Local error

Found 2 expressions with local error:

0.7b
(log1p (exp x))
0.0b
(- (log1p (exp x)) (* y x))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 32
Calls

2 calls. Slowest were:

6.0ms
(- (log1p (exp x)) (* y x))
1.0ms
(log1p (exp x))

series115.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

58.0ms
(- (log1p (exp x)) (* y x))
57.0ms
(log1p (exp x))

simplify199.0ms

Counts
17 → 38
Calls

17 calls. Slowest were:

35.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
32.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
20.0ms
(fma 1 (log1p (exp x)) (- (* x y)))

prune383.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

7.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
0.0ms
(fma x (fma 1/8 x 1/2) (log 2))

series53.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
18.0ms
(fma x (fma 1/8 x 1/2) (log 2))

simplify884.0ms

Counts
16 → 37
Calls

16 calls. Slowest were:

160.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
128.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
100.0ms
(fma (sqrt (fma x (fma 1/8 x 1/2) (log 2))) (sqrt (fma x (fma 1/8 x 1/2) (log 2))) (- (* x y)))

prune395.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 2 expressions with local error:

50.8b
(fma (- x) y (* x y))
0.0b
(fma x (- (fma 1/8 x 1/2) y) (log 2))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
2 → 20
Calls

2 calls. Slowest were:

0.0ms
(fma (- x) y (* x y))
0.0ms
(fma x (- (fma 1/8 x 1/2) y) (log 2))

series53.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(fma x (- (fma 1/8 x 1/2) y) (log 2))
18.0ms
(fma (- x) y (* x y))

simplify389.0ms

Counts
6 → 26
Calls

6 calls. Slowest were:

184.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
141.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
60.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

prune331.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

0.7b
(log1p (exp x))
0.7b
(log1p (exp x))
0.6b
(* (sqrt (- (log1p (exp x)) (* y x))) (sqrt (- (log1p (exp x)) (* y x))))
0.5b
(sqrt (- (log1p (exp x)) (* y x)))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 78
Calls

4 calls. Slowest were:

13.0ms
(* (sqrt (- (log1p (exp x)) (* y x))) (sqrt (- (log1p (exp x)) (* y x))))
7.0ms
(sqrt (- (log1p (exp x)) (* y x)))
0.0ms
(log1p (exp x))

series267.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

79.0ms
(log1p (exp x))
69.0ms
(sqrt (- (log1p (exp x)) (* y x)))
61.0ms
(log1p (exp x))
58.0ms
(* (sqrt (- (log1p (exp x)) (* y x))) (sqrt (- (log1p (exp x)) (* y x))))

simplify4.5s

Counts
55 → 90
Calls

55 calls. Slowest were:

718.0ms
(sqrt (- (* (log1p (exp x)) (log1p (exp x))) (* (* y x) (* y x))))
391.0ms
(* (sqrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3))) (sqrt (- (* (log1p (exp x)) (log1p (exp x))) (* (* y x) (* y x)))))
378.0ms
(* (sqrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3))) (sqrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3))))

prune852.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

regimes93.0ms

Accuracy

0% (0.2b remaining)

Error of 0.4b against oracle of 0.2b and baseline of 0.4b

bsearch1.0ms

end0.0ms

sample4.3s

Algorithm
intervals