Average Error: 39.3 → 0.1
Time: 13.4s
Precision: 64
Internal Precision: 128
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0001326945663731298:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.3
Target0.2
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.0001326945663731298

    1. Initial program 59.0

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

    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 0.0001326945663731298 < x

    1. Initial program 0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0001326945663731298:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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: 13.0s)Debug log

sample138.0ms

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: 39.5b

localize12.0ms

Local error

Found 1 expressions with local error:

4.3b
(log (+ 1 x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

2.0ms
(log (+ 1 x))

series53.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

53.0ms
(log (+ 1 x))

simplify638.0ms

Counts
9 → 17
Calls

9 calls. Slowest were:

353.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
164.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))
91.0ms
(- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))

prune178.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize30.0ms

Local error

Found 2 expressions with local error:

0.3b
(* x 1/3)
0.0b
(* (- (* 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)

series63.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

51.0ms
(* (- (* x 1/3) 1/2) (* x x))
12.0ms
(* x 1/3)

simplify1.5s

Counts
13 → 31
Calls

13 calls. Slowest were:

478.0ms
(* (- (* (* x 1/3) (* x 1/3)) (* 1/2 1/2)) (* x x))
453.0ms
(* (* (* (- (* x 1/3) 1/2) (- (* x 1/3) 1/2)) (- (* x 1/3) 1/2)) (* (* (* x x) (* x x)) (* x x)))
406.0ms
(* (- (pow (* x 1/3) 3) (pow 1/2 3)) (* x x))

prune214.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 4 expressions with local error:

4.3b
(log (+ 1 x))
4.3b
(log (+ 1 x))
4.3b
(log (+ 1 x))
0.2b
(cbrt (log (+ 1 x)))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

2.0ms
(log (+ 1 x))
2.0ms
(log (+ 1 x))
1.0ms
(log (+ 1 x))

series116.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

47.0ms
(cbrt (log (+ 1 x)))
25.0ms
(log (+ 1 x))
22.0ms
(log (+ 1 x))
22.0ms
(log (+ 1 x))

simplify2.6s

Counts
34 → 66
Calls

34 calls. Slowest were:

412.0ms
(- (+ (* 1/6 (* (pow (/ -1 (pow (log (/ 1 x)) 2)) 1/3) (/ 1 (pow x 2)))) (pow (- (log (/ 1 x))) 1/3)) (+ (* 1/3 (* (pow (/ -1 (pow (log (/ 1 x)) 2)) 1/3) (/ 1 x))) (* 1/9 (* (pow (/ -1 (pow (log (/ 1 x)) 5)) 1/3) (/ 1 (pow x 2))))))
370.0ms
(- (+ (* 1/3 (* (/ 1 x) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 2)) 1/3))) (pow (- (log -1) (log (/ -1 x))) 1/3)) (+ (* 1/6 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 2)) 1/3))) (* 1/9 (* (/ 1 (pow x 2)) (pow (/ 1 (pow (- (log -1) (log (/ -1 x))) 5)) 1/3)))))
338.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))

prune625.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize30.0ms

Local error

Found 4 expressions with local error:

4.4b
(log (sqrt (+ 1 x)))
4.4b
(log (sqrt (+ 1 x)))
4.3b
(log (+ 1 x))
4.3b
(log (+ 1 x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

3.0ms
(log (sqrt (+ 1 x)))
2.0ms
(log (sqrt (+ 1 x)))
2.0ms
(log (+ 1 x))

series162.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

55.0ms
(log (sqrt (+ 1 x)))
50.0ms
(log (sqrt (+ 1 x)))
34.0ms
(log (+ 1 x))
23.0ms
(log (+ 1 x))

simplify1.7s

Counts
42 → 76
Calls

42 calls. Slowest were:

331.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
327.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
187.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))

prune941.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes171.0ms

Accuracy

99.5% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 20.6b

bsearch35.0ms

end0.0ms

sample3.7s

Algorithm
intervals