Average Error: 39.0 → 0.2
Time: 11.1s
Precision: 64
Internal Precision: 128
$\log \left(1 + x\right)$
$\begin{array}{l} \mathbf{if}\;x \le 0.0001756469204950462:\\ \;\;\;\;x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}$

# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if x < 0.0001756469204950462

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

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

## if 0.0001756469204950462 < x

1. Initial program 0.0

$\log \left(1 + x\right)$
2. Using strategy rm

$\leadsto \log \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}$
4. Applied log-prod0.1

$\leadsto \color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}$
3. Recombined 2 regimes into one program.
4. Final simplification0.2

$\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0001756469204950462:\\ \;\;\;\;x - \left(x \cdot x\right) \cdot \left(\frac{1}{2} - \frac{1}{3} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}$

# Reproduce

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)'
(FPCore (x)
:name "log(1+x)"
(log (+ 1 x)))

# Details

Time bar (total: 5.6s)Debug log

Algorithm
intervals

### setup61.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 40.7b

### localize20.0ms

Local error

Found 1 expressions with local error:

 4.1b (log (+ 1 x))

### rewrite8.0ms

Algorithm
Counts
1 → 14
Calls

1 calls. Slowest were:

 5.0ms (log (+ 1 x))

### series60.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

 59.0ms (log (+ 1 x))

### simplify1.1s

Counts
9 → 17
Calls

9 calls. Slowest were:

 540.0ms (- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2))))) 273.0ms (- (/ 1 x) (+ (* 1/2 (/ 1 (pow x 2))) (log (/ 1 x)))) 203.0ms (- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))

### prune199.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

### localize29.0ms

Local error

Found 3 expressions with local error:

 0.3b (* 1/3 x) 0.0b (* (* x x) (- 1/2 (* 1/3 x))) 0.0b (- x (* (* x x) (- 1/2 (* 1/3 x))))

### rewrite13.0ms

Algorithm
Counts
3 → 44
Calls

3 calls. Slowest were:

 9.0ms (* (* x x) (- 1/2 (* 1/3 x))) 3.0ms (- x (* (* x x) (- 1/2 (* 1/3 x)))) 1.0ms (* 1/3 x)

### series81.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

 48.0ms (* (* x x) (- 1/2 (* 1/3 x))) 20.0ms (- x (* (* x x) (- 1/2 (* 1/3 x)))) 12.0ms (* 1/3 x)

### simplify3.1s

Counts
25 → 53
Calls

25 calls. Slowest were:

 1.3s (* (* (* (* x x) (* x x)) (* x x)) (* (* (- 1/2 (* 1/3 x)) (- 1/2 (* 1/3 x))) (- 1/2 (* 1/3 x)))) 199.0ms (- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2))) 195.0ms (- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))

### prune659.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

### regimes86.0ms

Accuracy

99.5% (0.1b remaining)

Error of 0.2b against oracle of 0.1b and baseline of 20.7b