Average Error: 38.8 → 0.2
Time: 14.5s
Precision: 64
Internal Precision: 128
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0001385879234543181:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.0001385879234543181

    1. Initial program 58.8

      \[\log \left(1 + x\right)\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot {x}^{3}\right) - \frac{1}{2} \cdot {x}^{2}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{3} - \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x}\]

    if 0.0001385879234543181 < x

    1. Initial program 0.1

      \[\log \left(1 + x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0001385879234543181:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))

Details

Time bar (total: 13.4s)Debug log

sample131.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

3.0ms
(log (+ 1 x))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 38.8b

localize12.0ms

Local error

Found 1 expressions with local error:

4.5b
(log (+ 1 x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

2.0ms
(log (+ 1 x))

series36.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

36.0ms
(log (+ 1 x))

simplify630.0ms

Counts
9 → 17
Calls

9 calls. Slowest were:

337.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
159.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))
112.0ms
(- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))

prune186.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.1b

localize26.0ms

Local error

Found 2 expressions with local error:

0.3b
(* x 1/3)
0.1b
(* (- (* x 1/3) 1/2) (* x x))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

11.0ms
(* (- (* x 1/3) 1/2) (* x x))
0.0ms
(* x 1/3)

series70.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

54.0ms
(* (- (* x 1/3) 1/2) (* x x))
16.0ms
(* x 1/3)

simplify1.4s

Counts
13 → 31
Calls

13 calls. Slowest were:

425.0ms
(* (- (* (* x 1/3) (* x 1/3)) (* 1/2 1/2)) (* x x))
386.0ms
(* (* (* (- (* x 1/3) 1/2) (- (* x 1/3) 1/2)) (- (* x 1/3) 1/2)) (* (* (* x x) (* x x)) (* x x)))
332.0ms
(* (- (pow (* x 1/3) 3) (pow 1/2 3)) (* x x))

prune228.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.1b

localize18.0ms

Local error

Found 4 expressions with local error:

4.5b
(log (+ 1 x))
4.5b
(log (+ 1 x))
4.5b
(log (+ 1 x))
0.2b
(* (cbrt (log (+ 1 x))) (cbrt (log (+ 1 x))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

10.0ms
(* (cbrt (log (+ 1 x))) (cbrt (log (+ 1 x))))
2.0ms
(log (+ 1 x))
2.0ms
(log (+ 1 x))

series171.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(* (cbrt (log (+ 1 x))) (cbrt (log (+ 1 x))))
33.0ms
(log (+ 1 x))
30.0ms
(log (+ 1 x))
23.0ms
(log (+ 1 x))

simplify2.4s

Counts
52 → 84
Calls

52 calls. Slowest were:

318.0ms
(- (+ (* 2/3 (* (/ (log -1) x) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 4)) 1/3))) (+ (* 8/9 (* (/ (* (log (/ -1 x)) (log -1)) (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 10)) 1/3))) (+ (* 1/3 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 4)) 1/3))) (+ (* 1/3 (* (/ (log (/ -1 x)) (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 4)) 1/3))) (pow (pow (- (log -1) (log (/ -1 x))) 2) 1/3))))) (+ (* 2/3 (* (/ (log (/ -1 x)) x) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 4)) 1/3))) (+ (* 1/3 (* (/ (log -1) (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 4)) 1/3))) (+ (* 4/9 (* (/ (pow (log -1) 2) (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 10)) 1/3))) (* 4/9 (* (/ (pow (log (/ -1 x)) 2) (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 10)) 1/3)))))))
302.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
300.0ms
(- (+ (* 1/3 (* (pow (/ 1 (log (/ 1 x))) 1/3) (/ 1 (pow x 2)))) (pow (pow (log (/ 1 x)) 2) 1/3)) (+ (* 1/9 (* (pow (/ 1 (pow (log (/ 1 x)) 4)) 1/3) (/ 1 (pow x 2)))) (* 2/3 (* (pow (/ 1 (log (/ 1 x))) 1/3) (/ 1 x)))))

prune918.0ms

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.1b

localize39.0ms

Local error

Found 4 expressions with local error:

4.5b
(log (+ 1 x))
4.5b
(log (+ 1 x))
4.4b
(log (cbrt (+ 1 x)))
4.3b
(log (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

12.0ms
(log (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))
3.0ms
(log (cbrt (+ 1 x)))
2.0ms
(log (+ 1 x))

series170.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

62.0ms
(log (* (cbrt (+ 1 x)) (cbrt (+ 1 x))))
50.0ms
(log (cbrt (+ 1 x)))
35.0ms
(log (+ 1 x))
23.0ms
(log (+ 1 x))

simplify2.1s

Counts
47 → 83
Calls

47 calls. Slowest were:

315.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
296.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
292.0ms
(- (+ (* 1/3 (/ 1 x)) (* 1/3 (log -1))) (+ (* 1/3 (log (/ -1 x))) (* 1/6 (/ 1 (pow x 2)))))

prune1.1s

Pruning

6 alts after pruning (3 fresh and 3 done)

Merged error: 0.1b

regimes243.0ms

Accuracy

99.5% (0.1b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 21.0b

bsearch43.0ms

end0.0ms

sample3.5s

Algorithm
intervals