Average Error: 0.5 → 0.4
Time: 23.4s
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 2019007 +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: 22.7s)Debug log

sample105.0ms

Algorithm
intervals

simplify18.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune35.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize13.0ms

Local error

Found 2 expressions with local error:

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

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
2 → 32
Calls

2 calls. Slowest were:

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

series116.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

66.0ms
(- (log1p (exp x)) (* y x))
50.0ms
(log1p (exp x))

simplify225.0ms

Counts
17 → 38
Calls

17 calls. Slowest were:

44.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
36.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
21.0ms
(fma (* (cbrt (log1p (exp x))) (cbrt (log1p (exp x)))) (cbrt (log1p (exp x))) (- (* x y)))

prune428.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.4b

localize26.0ms

Local error

Found 2 expressions with local error:

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

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))

series62.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

35.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
27.0ms
(fma x (fma 1/8 x 1/2) (log 2))

simplify980.0ms

Counts
16 → 37
Calls

16 calls. Slowest were:

162.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
138.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
127.0ms
(/ (exp (fma x (fma 1/8 x 1/2) (log 2))) (exp (* y x)))

prune387.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.4b

localize26.0ms

Local error

Found 2 expressions with local error:

50.1b
(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))

series41.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

25.0ms
(fma x (- (fma 1/8 x 1/2) y) (log 2))
16.0ms
(fma (- x) y (* x y))

simplify398.0ms

Counts
6 → 26
Calls

6 calls. Slowest were:

179.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
171.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
45.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))

prune330.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.4b

localize48.0ms

Local error

Found 4 expressions with local error:

15.5b
(/ (- (pow (log1p (exp x)) 3) (pow (* y x) 3)) (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x)))))
0.7b
(* (log1p (exp x)) (log1p (exp x)))
0.7b
(pow (log1p (exp x)) 3)
0.5b
(log1p (exp x))

rewrite76.0ms

Algorithm
rewrite-expression-head
Counts
4 → 108
Calls

4 calls. Slowest were:

63.0ms
(/ (- (pow (log1p (exp x)) 3) (pow (* y x) 3)) (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x)))))
8.0ms
(* (log1p (exp x)) (log1p (exp x)))
2.0ms
(pow (log1p (exp x)) 3)

series420.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

158.0ms
(/ (- (pow (log1p (exp x)) 3) (pow (* y x) 3)) (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x)))))
112.0ms
(pow (log1p (exp x)) 3)
91.0ms
(* (log1p (exp x)) (log1p (exp x)))
59.0ms
(log1p (exp x))

simplify12.7s

Counts
97 → 120
Calls

97 calls. Slowest were:

446.0ms
(/ (* (cbrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3))) (cbrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3)))) (* (cbrt (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x))))) (cbrt (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x)))))))
373.0ms
(/ (sqrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3))) (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x)))))
369.0ms
(/ (cbrt (- (pow (log1p (exp x)) 3) (pow (* y x) 3))) (cbrt (+ (* (log1p (exp x)) (log1p (exp x))) (+ (* (* y x) (* y x)) (* (log1p (exp x)) (* y x))))))

prune1.8s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.4b

regimes90.0ms

Accuracy

0% (0.2b remaining)

Error of 0.4b against oracle of 0.3b and baseline of 0.4b

bsearch1.0ms

end0.0ms

sample4.5s

Algorithm
intervals