Average Error: 0.5 → 0.5
Time: 14.9s
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
Target0.0
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. Simplified0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019005 +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: 14.1s)Debug log

sample124.0ms

Algorithm
intervals

simplify21.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

21.0ms
(- (log (+ 1 (exp x))) (* x y))

prune16.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.7b

localize19.0ms

Local error

Found 2 expressions with local error:

0.7b
(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))

series123.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

63.0ms
(log1p (exp x))
60.0ms
(- (log1p (exp x)) (* y x))

simplify264.0ms

Counts
17 → 38
Calls

17 calls. Slowest were:

50.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
50.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
22.0ms
(fma 1 (log1p (exp x)) (- (* x y)))

prune322.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.4b

localize13.0ms

Local error

Found 4 expressions with local error:

1.0b
(expm1 (log1p (- (log1p (exp x)) (* y x))))
0.7b
(log1p (exp x))
0.0b
(log1p (- (log1p (exp x)) (* y x)))
0.0b
(- (log1p (exp x)) (* y x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

7.0ms
(- (log1p (exp x)) (* y x))
1.0ms
(log1p (- (log1p (exp x)) (* y x)))
1.0ms
(log1p (exp x))

series579.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

287.0ms
(expm1 (log1p (- (log1p (exp x)) (* y x))))
188.0ms
(log1p (- (log1p (exp x)) (* y x)))
58.0ms
(- (log1p (exp x)) (* y x))
46.0ms
(log1p (exp x))

simplify1.7s

Counts
25 → 66
Calls

25 calls. Slowest were:

381.0ms
(- (exp (- (log -1) (+ (log (/ -1 x)) (log (/ -1 y))))) (+ (/ (exp (- (log -1) (+ (log (/ -1 x)) (log (/ -1 y))))) (* x y)) (+ 1 (/ (* (exp (- (log -1) (+ (log (/ -1 x)) (log (/ -1 y))))) (log (+ (exp x) 1))) (* x y)))))
253.0ms
(+ (log 2) (+ (* 1/8 (pow x 2)) (+ (* 1/2 (/ (* (log 2) x) (+ (log 2) 1))) (* 1/2 (/ x (+ (log 2) 1))))))
231.0ms
(- (log -1) (+ (* 1/2 (/ (pow (log (+ (exp x) 1)) 2) (* (pow x 2) (pow y 2)))) (+ (* 1/2 (/ 1 (* (pow x 2) (pow y 2)))) (+ (/ (log (+ (exp x) 1)) (* x y)) (+ (log (/ -1 x)) (+ (log (/ -1 y)) (+ (/ 1 (* x y)) (/ (log (+ (exp x) 1)) (* (pow x 2) (pow y 2))))))))))

prune673.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.4b

localize18.0ms

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))

series68.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

43.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
24.0ms
(fma x (fma 1/8 x 1/2) (log 2))

simplify928.0ms

Counts
16 → 37
Calls

16 calls. Slowest were:

166.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
129.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
118.0ms
(/ (exp (fma x (fma 1/8 x 1/2) (log 2))) (exp (* y x)))

prune386.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.4b

localize9.0ms

Local error

Found 4 expressions with local error:

5.1b
(log (fma x (fma 1/8 x 1/2) (log 2)))
1.0b
(exp (log (fma x (fma 1/8 x 1/2) (log 2))))
0.0b
(- (exp (log (fma x (fma 1/8 x 1/2) (log 2)))) (* y x))
0.0b
(fma x (fma 1/8 x 1/2) (log 2))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

11.0ms
(- (exp (log (fma x (fma 1/8 x 1/2) (log 2)))) (* y x))
2.0ms
(exp (log (fma x (fma 1/8 x 1/2) (log 2))))
1.0ms
(log (fma x (fma 1/8 x 1/2) (log 2)))

series106.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

39.0ms
(log (fma x (fma 1/8 x 1/2) (log 2)))
27.0ms
(- (exp (log (fma x (fma 1/8 x 1/2) (log 2)))) (* y x))
23.0ms
(fma x (fma 1/8 x 1/2) (log 2))
16.0ms
(exp (log (fma x (fma 1/8 x 1/2) (log 2))))

simplify3.3s

Counts
41 → 79
Calls

41 calls. Slowest were:

467.0ms
(- (+ (* 1/8 (/ (pow x 2) (log 2))) (+ (log (log 2)) (* 1/2 (/ x (log 2))))) (* 1/8 (/ (pow x 2) (pow (log 2) 2))))
316.0ms
(- (+ (log 1/8) (+ (* 8 (/ (log 2) (pow x 2))) (* 4 (/ 1 x)))) (+ (* 2 (log (/ -1 x))) (* 8 (/ 1 (pow x 2)))))
303.0ms
(/ (exp (exp (log (fma x (fma 1/8 x 1/2) (log 2))))) (exp (* y x)))

prune908.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.4b

regimes95.0ms

Accuracy

0% (0.3b remaining)

Error of 0.5b against oracle of 0.2b and baseline of 0.5b

bsearch1.0ms

end0.0ms

sample4.4s

Algorithm
intervals