Average Error: 39.7 → 0.2
Time: 14.8s
Precision: 64
Internal Precision: 128
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.00011717543999484213:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.7
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 < 0.00011717543999484213

    1. Initial program 58.7

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

    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 0.00011717543999484213 < x

    1. Initial program 0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00011717543999484213:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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: 10.5s)Debug log

start135.0ms

Algorithm
intervals

setup27.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.2b

localize13.0ms

Local error

Found 1 expressions with local error:

5.3b
(log (+ 1 x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
1 → 14
Calls

1 calls. Slowest were:

2.0ms
(log (+ 1 x))

series25.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

25.0ms
(log (+ 1 x))

simplify594.0ms

Counts
9 → 17
Calls

9 calls. Slowest were:

347.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
135.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))
97.0ms
(- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))

prune200.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 2 expressions with local error:

0.4b
(* x 1/3)
0.0b
(* (- (* x 1/3) 1/2) (* x x))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
2 → 25
Calls

2 calls. Slowest were:

6.0ms
(* (- (* x 1/3) 1/2) (* x x))
0.0ms
(* x 1/3)

series40.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

33.0ms
(* (- (* x 1/3) 1/2) (* x x))
8.0ms
(* x 1/3)

simplify1.6s

Counts
13 → 31
Calls

13 calls. Slowest were:

539.0ms
(* (- (* (* x 1/3) (* x 1/3)) (* 1/2 1/2)) (* x x))
457.0ms
(* (- (pow (* x 1/3) 3) (pow 1/2 3)) (* x x))
444.0ms
(* (* (* (- (* x 1/3) 1/2) (- (* x 1/3) 1/2)) (- (* x 1/3) 1/2)) (* (* (* x x) (* x x)) (* x x)))

prune347.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 4 expressions with local error:

5.3b
(log (+ 1 x))
5.3b
(log (+ 1 x))
5.3b
(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))
2.0ms
(log (+ 1 x))

series127.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

48.0ms
(cbrt (log (+ 1 x)))
30.0ms
(log (+ 1 x))
27.0ms
(log (+ 1 x))
22.0ms
(log (+ 1 x))

simplify2.6s

Counts
34 → 66
Calls

34 calls. Slowest were:

387.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
372.0ms
(- (+ (* 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))))))
344.0ms
(- (+ (* 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)))))

prune704.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

5.3b
(log (- 1 x))
5.3b
(log (+ 1 x))
5.3b
(log (+ 1 x))
2.2b
(log (- 1 (* x x)))

rewrite4.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

1.0ms
(log (- 1 x))
1.0ms
(log (- 1 (* x x)))
1.0ms
(log (+ 1 x))

series161.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

84.0ms
(log (- 1 (* x x)))
32.0ms
(log (- 1 x))
23.0ms
(log (+ 1 x))
22.0ms
(log (+ 1 x))

simplify2.6s

Counts
36 → 68
Calls

36 calls. Slowest were:

394.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
306.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
238.0ms
(- (log -1) (+ (log (/ 1 x)) (+ (* 1/2 (/ 1 (pow x 2))) (/ 1 x))))

prune903.0ms

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0.0b

regimes217.0ms

Accuracy

99.3% (0.1b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 19.9b

bsearch59.0ms