Average Error: 0.5 → 0.4
Time: 9.9s
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. Initial simplification0.4

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

  3. Taylor expanded around -inf 0.5

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2018360 +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: 4.3s)Debug log

start113.0ms

Algorithm
intervals

setup24.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 2 expressions with local error:

0.5b
(log1p (exp x))
0.0b
(- (log1p (exp x)) (* y x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 32
Calls

2 calls. Slowest were:

3.0ms
(- (log1p (exp x)) (* y x))
0.0ms
(log1p (exp x))

series126.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

75.0ms
(log1p (exp x))
51.0ms
(- (log1p (exp x)) (* y x))

simplify232.0ms

Counts
17 → 38
Calls

17 calls. Slowest were:

52.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
34.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
20.0ms
(fma (* (cbrt (log1p (exp x))) (cbrt (log1p (exp x)))) (cbrt (log1p (exp x))) (- (* x y)))

prune390.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.2b

localize1.0ms

Local error

Found 2 expressions with local error:

0.5b
(log1p (exp x))
0.0b
(- (log1p (exp x)) (* y x))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
2 → 32
Calls

2 calls. Slowest were:

4.0ms
(- (log1p (exp x)) (* y x))
0.0ms
(log1p (exp x))

series113.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

66.0ms
(log1p (exp x))
47.0ms
(- (log1p (exp x)) (* y x))

simplify171.0ms

Counts
17 → 38
Calls

17 calls. Slowest were:

30.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
26.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
19.0ms
(fma 1 (log1p (exp x)) (- (* x y)))

prune435.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.2b

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))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 31
Calls

2 calls. Slowest were:

6.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
0.0ms
(fma x (fma 1/8 x 1/2) (log 2))

series73.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

42.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
31.0ms
(fma x (fma 1/8 x 1/2) (log 2))

simplify877.0ms

Counts
16 → 37
Calls

16 calls. Slowest were:

173.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
125.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
120.0ms
(/ (exp (fma x (fma 1/8 x 1/2) (log 2))) (exp (* y x)))

prune576.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.2b

localize27.0ms

Local error

Found 2 expressions with local error:

50.2b
(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))

series45.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

30.0ms
(fma x (- (fma 1/8 x 1/2) y) (log 2))
15.0ms
(fma (- x) y (* x y))

simplify422.0ms

Counts
6 → 26
Calls

6 calls. Slowest were:

188.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
185.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
45.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

prune481.0ms

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.2b

regimes134.0ms

Accuracy

0% (0.2b remaining)

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

bsearch2.0ms