Average Error: 0.5 → 0.5
Time: 34.6s
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 2019016 +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: 34.0s)Debug log

sample151.0ms

Algorithm
intervals
Results
359×(pre true 80)
262×(body real 80)
71×(body exit 10240)
12×(body real 640)
9×(body real 320)
3×(body real 160)
2×(body real 1280)

simplify12.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
12.0ms
(- (log (+ 1 (exp x))) (* x y))

prune19.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

Local error

Found 2 expressions with local error:

0.3b
(log1p (exp x))
0.0b
(- (log1p (exp x)) (* y x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×prod-diff
3×fma-neg
3×expm1-log1p-u
2×log1p-expm1-u
2×add-exp-log
2×diff-log
2×add-cbrt-cube
2×log1p-udef
2×pow1
1×flip--
1×log1p-expm1
1×flip3--
1×sub-neg
Counts
2 → 32
Calls
2 calls:
Slowest
4.0ms
(- (log1p (exp x)) (* y x))
0.0ms
(log1p (exp x))

series118.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
60.0ms
(log1p (exp x))
58.0ms
(- (log1p (exp x)) (* y x))

simplify198.0ms

Counts
17 → 38
Calls
17 calls:
Slowest
54.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
34.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
19.0ms
(fma (* (cbrt (log1p (exp x))) (cbrt (log1p (exp x)))) (cbrt (log1p (exp x))) (- (* x y)))
15.0ms
(- (log (+ (exp x) 1)) (* x y))
13.0ms
(/ (exp (log1p (exp x))) (exp (* y x)))

prune373.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.2b

localize12.0ms

Local error

Found 4 expressions with local error:

15.7b
(cbrt (* (* (- (log1p (exp x)) (* y x)) (- (log1p (exp x)) (* y x))) (- (log1p (exp x)) (* y x))))
0.6b
(* (- (log1p (exp x)) (* y x)) (- (log1p (exp x)) (* y x)))
0.3b
(log1p (exp x))
0.3b
(log1p (exp x))

rewrite39.0ms

Algorithm
rewrite-expression-head
Rules
33×flip--
33×flip3--
32×frac-times
26×cbrt-div
16×associate-*l/
10×associate-*r/
9×pow1
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×add-exp-log
6×prod-diff
6×add-cbrt-cube
6×expm1-log1p-u
5×distribute-lft-in
5×distribute-rgt-in
4×add-log-exp
4×log1p-expm1-u
3×associate-*l*
3×associate-*r*
2×log1p-expm1
2×sub-neg
2×log1p-udef
1×cbrt-unprod
1×rem-cbrt-cube
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-prod-down
1×pow3
1×pow1/3
1×pow-plus
1×pow2
1×cbrt-prod
Counts
4 → 100
Calls
4 calls:
Slowest
22.0ms
(cbrt (* (* (- (log1p (exp x)) (* y x)) (- (log1p (exp x)) (* y x))) (- (log1p (exp x)) (* y x))))
15.0ms
(* (- (log1p (exp x)) (* y x)) (- (log1p (exp x)) (* y x)))
0.0ms
(log1p (exp x))
0.0ms
(log1p (exp x))

series264.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
82.0ms
(* (- (log1p (exp x)) (* y x)) (- (log1p (exp x)) (* y x)))
70.0ms
(log1p (exp x))
68.0ms
(log1p (exp x))
44.0ms
(cbrt (* (* (- (log1p (exp x)) (* y x)) (- (log1p (exp x)) (* y x))) (- (log1p (exp x)) (* y x))))

simplify23.7s

Counts
75 → 112
Calls
75 calls:
Slowest
1.4s
(* (fma (- x) y (* x y)) (- (log1p (exp x)) (* y x)))
1.3s
(cbrt (* (* (- (* (log1p (exp x)) (log1p (exp x))) (* (* y x) (* y x))) (- (log1p (exp x)) (* y x))) (- (* (log1p (exp x)) (log1p (exp x))) (* (* y x) (* y x)))))
1.3s
(* (fma (- x) y (* x y)) (- (log1p (exp x)) (* y x)))
1.3s
(* (fma (- x) y (* x y)) (- (log1p (exp x)) (* y x)))
1.2s
(* (- (log1p (exp x)) (* y x)) (fma (- x) y (* x y)))

prune1.3s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.2b

localize30.0ms

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))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×prod-diff
3×fma-neg
2×log1p-expm1-u
2×fma-udef
2×add-exp-log
2×add-cbrt-cube
2×pow1
2×expm1-log1p-u
1×flip--
1×associate--l+
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 31
Calls
2 calls:
Slowest
8.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
0.0ms
(fma x (fma 1/8 x 1/2) (log 2))

series54.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
28.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
26.0ms
(fma x (fma 1/8 x 1/2) (log 2))

simplify1.0s

Counts
16 → 37
Calls
16 calls:
Slowest
187.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
175.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
118.0ms
(fma 1 (fma x (fma 1/8 x 1/2) (log 2)) (- (* x y)))
91.0ms
(fma (* (cbrt (fma x (fma 1/8 x 1/2) (log 2))) (cbrt (fma x (fma 1/8 x 1/2) (log 2)))) (cbrt (fma x (fma 1/8 x 1/2) (log 2))) (- (* x y)))
89.0ms
(/ (exp (fma x (fma 1/8 x 1/2) (log 2))) (exp (* y x)))

prune349.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.2b

localize1.0ms

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
Rules
4×add-log-exp
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×prod-diff
3×fma-neg
2×log1p-expm1-u
2×fma-udef
2×add-exp-log
2×add-cbrt-cube
2×pow1
2×expm1-log1p-u
1×flip--
1×associate--l+
1×diff-log
1×flip3--
1×sub-neg
Counts
2 → 31
Calls
2 calls:
Slowest
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))

series66.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
43.0ms
(- (fma x (fma 1/8 x 1/2) (log 2)) (* y x))
23.0ms
(fma x (fma 1/8 x 1/2) (log 2))

simplify957.0ms

Counts
16 → 37
Calls
16 calls:
Slowest
139.0ms
(/ (exp (fma x (fma 1/8 x 1/2) (log 2))) (exp (* y x)))
129.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
120.0ms
(- (+ (* 1/2 x) (* 1/8 (pow x 2))) (* x y))
116.0ms
(fma 1 (fma x (fma 1/8 x 1/2) (log 2)) (- (* x y)))
113.0ms
(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)))

prune386.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.2b

regimes77.0ms

Accuracy

0% (0.2b remaining)

Error of 0.5b against oracle of 0.3b and baseline of 0.5b

bsearch2.0ms

end0.0ms

sample4.8s

Algorithm
intervals
Results
11544×(pre true 80)
8434×(body real 80)
2466×(body exit 10240)
256×(body real 640)
165×(body real 320)
122×(body real 1280)
101×(body real 160)