Average Error: 0.6 → 0.6
Time: 29.3s
Precision: 64
Internal Precision: 128
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left({\left(e^{x}\right)}^{3} + 1\right) - \left(y \cdot x + \log \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right)\right)\]

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.6

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  2. Using strategy rm

  3. Applied flip3-+0.6

    \[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)} - x \cdot y\]

  4. Applied log-div0.6

    \[\leadsto \color{blue}{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right)} - x \cdot y\]

  5. Applied associate--l-0.6

    \[\leadsto \color{blue}{\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + x \cdot y\right)}\]

  6. Simplified0.6

    \[\leadsto \color{blue}{\log \left({\left(e^{x}\right)}^{3} + 1\right)} - \left(\log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right) + x \cdot y\right)\]

  7. Final simplification0.6

    \[\leadsto \log \left({\left(e^{x}\right)}^{3} + 1\right) - \left(y \cdot x + \log \left(1 + \left(e^{x} \cdot e^{x} - e^{x}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019010 
(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: 28.6s)Debug log

sample137.0ms

Algorithm
intervals

simplify6.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
6.0ms
(- (log (+ 1 (exp x))) (* x y))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize15.0ms

Local error

Found 3 expressions with local error:

0.5b
(log (+ 1 (exp x)))
0.0b
(- (log (+ 1 (exp x))) (* x y))
0.0b
(+ 1 (exp x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
6×log-prod
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×add-exp-log
4×log-div
4×pow1
3×flip-+
3×associate--l+
3×add-cbrt-cube
3×flip3-+
2×associate--l-
1×flip--
1×log-pow
1×rem-log-exp
1×diff-log
1×flip3--
1×sub-neg
1×+-commutative
Counts
3 → 41
Calls
3 calls:
Slowest
5.0ms
(- (log (+ 1 (exp x))) (* x y))
2.0ms
(log (+ 1 (exp x)))
1.0ms
(+ 1 (exp x))

series116.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
53.0ms
(log (+ 1 (exp x)))
52.0ms
(- (log (+ 1 (exp x))) (* x y))
11.0ms
(+ 1 (exp x))

simplify209.0ms

Counts
21 → 50
Calls
21 calls:
Slowest
38.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
35.0ms
(+ (log 2) (+ (* 1/2 x) (* 1/8 (pow x 2))))
32.0ms
(+ x (+ (* 1/2 (pow x 2)) 2))
16.0ms
(log (+ (pow 1 3) (pow (exp x) 3)))
15.0ms
(log (+ (pow 1 3) (pow (exp x) 3)))

prune454.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.2b

localize40.0ms

Local error

Found 4 expressions with local error:

4.5b
(log (+ (* 1 1) (- (* (exp x) (exp x)) (* 1 (exp x)))))
4.4b
(- (* (exp x) (exp x)) (* 1 (exp x)))
0.5b
(log (+ (pow (exp x) 3) 1))
0.0b
(pow (exp x) 3)

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
9×*-un-lft-identity
8×add-cube-cbrt
8×add-sqr-sqrt
7×log-prod
7×pow1
6×add-log-exp
6×add-exp-log
5×add-cbrt-cube
4×log-div
3×cube-prod
3×unpow-prod-down
2×flip-+
2×log-pow
2×rem-log-exp
2×flip3-+
1×rem-cube-cbrt
1×distribute-lft-out
1×flip--
1×pow-exp
1×unpow3
1×diff-log
1×distribute-rgt-out--
1×flip3--
1×pow-pow
1×sub-neg
1×pow-to-exp
1×cube-mult
Counts
4 → 61
Calls
4 calls:
Slowest
6.0ms
(log (+ (* 1 1) (- (* (exp x) (exp x)) (* 1 (exp x)))))
5.0ms
(- (* (exp x) (exp x)) (* 1 (exp x)))
3.0ms
(log (+ (pow (exp x) 3) 1))
1.0ms
(pow (exp x) 3)

series225.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
102.0ms
(log (+ (* 1 1) (- (* (exp x) (exp x)) (* 1 (exp x)))))
83.0ms
(log (+ (pow (exp x) 3) 1))
26.0ms
(- (* (exp x) (exp x)) (* 1 (exp x)))
14.0ms
(pow (exp x) 3)

simplify1.4s

Counts
37 → 73
Calls
37 calls:
Slowest
339.0ms
(log (- (* (* 1 1) (* 1 1)) (* (- (* (exp x) (exp x)) (* 1 (exp x))) (- (* (exp x) (exp x)) (* 1 (exp x))))))
183.0ms
(log (+ (pow (* 1 1) 3) (pow (- (* (exp x) (exp x)) (* 1 (exp x))) 3)))
118.0ms
(log (+ (pow (pow (exp x) 3) 3) (pow 1 3)))
105.0ms
(log (sqrt (+ (* 1 1) (- (* (exp x) (exp x)) (* 1 (exp x))))))
89.0ms
(log (- (* (pow (exp x) 3) (pow (exp x) 3)) (* 1 1)))

prune1.0s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.2b

localize19.0ms

Local error

Found 4 expressions with local error:

14.7b
(cbrt (* (* (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp x))) (* x y))) (- (log (+ 1 (exp x))) (* x y))))
0.6b
(* (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp x))) (* x y)))
0.5b
(log (+ 1 (exp x)))
0.5b
(log (+ 1 (exp x)))

rewrite84.0ms

Algorithm
rewrite-expression-head
Rules
33×flip--
33×flip3--
32×frac-times
26×cbrt-div
16×associate-*l/
12×log-prod
11×pow1
10×add-cube-cbrt
10×associate-*r/
10×*-un-lft-identity
10×add-sqr-sqrt
8×add-exp-log
6×associate--l+
6×add-cbrt-cube
5×distribute-lft-in
5×distribute-rgt-in
4×add-log-exp
4×log-div
3×associate-*l*
3×associate-*r*
2×flip-+
2×log-pow
2×rem-log-exp
2×flip3-+
2×sub-neg
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 → 102
Calls
4 calls:
Slowest
42.0ms
(cbrt (* (* (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp x))) (* x y))) (- (log (+ 1 (exp x))) (* x y))))
34.0ms
(* (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp x))) (* x y)))
2.0ms
(log (+ 1 (exp x)))
2.0ms
(log (+ 1 (exp x)))

series239.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
74.0ms
(log (+ 1 (exp x)))
62.0ms
(log (+ 1 (exp x)))
60.0ms
(* (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp x))) (* x y)))
43.0ms
(cbrt (* (* (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp x))) (* x y))) (- (log (+ 1 (exp x))) (* x y))))

simplify15.6s

Counts
85 → 114
Calls
85 calls:
Slowest
966.0ms
(cbrt (* (* (- (* (log (+ 1 (exp x))) (log (+ 1 (exp x)))) (* (* x y) (* x y))) (- (* (log (+ 1 (exp x))) (log (+ 1 (exp x)))) (* (* x y) (* x y)))) (- (log (+ 1 (exp x))) (* x y))))
897.0ms
(cbrt (* (* (- (pow (log (+ 1 (exp x))) 3) (pow (* x y) 3)) (- (pow (log (+ 1 (exp x))) 3) (pow (* x y) 3))) (- (* (log (+ 1 (exp x))) (log (+ 1 (exp x)))) (* (* x y) (* x y)))))
845.0ms
(cbrt (* (* (- (* (log (+ 1 (exp x))) (log (+ 1 (exp x)))) (* (* x y) (* x y))) (- (log (+ 1 (exp x))) (* x y))) (- (* (log (+ 1 (exp x))) (log (+ 1 (exp x)))) (* (* x y) (* x y)))))
608.0ms
(cbrt (* (* (- (log (+ 1 (exp x))) (* x y)) (- (pow (log (+ 1 (exp x))) 3) (pow (* x y) 3))) (- (pow (log (+ 1 (exp x))) 3) (pow (* x y) 3))))
563.0ms
(cbrt (* (* (- (log (+ 1 (exp x))) (* x y)) (- (* (log (+ 1 (exp x))) (log (+ 1 (exp x)))) (* (* x y) (* x y)))) (- (log (+ 1 (exp x))) (* x y))))

prune1.5s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.2b

localize36.0ms

Local error

Found 4 expressions with local error:

9.9b
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
0.0b
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
0.0b
(+ (* 1/2 (pow x 2)) 2)
0.0b
(* 1/2 (pow x 2))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×log-prod
5×add-log-exp
5×add-exp-log
5×pow1
4×associate-*r*
4×add-cbrt-cube
4×log-div
3×flip-+
3×associate--l+
3×flip3-+
2×associate--l-
1×unpow2
1×flip--
1×*-commutative
1×log-pow
1×rem-log-exp
1×diff-log
1×flip3--
1×sub-neg
1×+-commutative
Counts
4 → 53
Calls
4 calls:
Slowest
10.0ms
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
4.0ms
(+ (* 1/2 (pow x 2)) 2)
3.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
2.0ms
(* 1/2 (pow x 2))

series121.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
48.0ms
(- (log (+ x (+ (* 1/2 (pow x 2)) 2))) (* x y))
30.0ms
(log (+ x (+ (* 1/2 (pow x 2)) 2)))
24.0ms
(+ (* 1/2 (pow x 2)) 2)
19.0ms
(* 1/2 (pow x 2))

simplify1.7s

Counts
27 → 65
Calls
27 calls:
Slowest
301.0ms
(- (+ (log 1/2) (+ (* 2 (/ 1 (pow x 2))) (* 2 (/ 1 x)))) (* 2 (log (/ 1 x))))
183.0ms
(- (+ (log 1/2) (* 2 (/ 1 x))) (+ (* 2 (log (/ 1 x))) (* x y)))
168.0ms
(log (- (* x x) (* (+ (* 1/2 (pow x 2)) 2) (+ (* 1/2 (pow x 2)) 2))))
164.0ms
(- (+ (log 1/2) (* 2 (/ 1 x))) (+ (* 2 (log (/ -1 x))) (* x y)))
143.0ms
(log (+ (pow x 3) (pow (+ (* 1/2 (pow x 2)) 2) 3)))

prune679.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.2b

regimes152.0ms

Accuracy

0% (0.3b remaining)

Error of 0.6b against oracle of 0.3b and baseline of 0.6b

bsearch2.0ms

end0.0ms

sample4.8s

Algorithm
intervals