Average Error: 39.2 → 0.2
Time: 12.9s
Precision: 64
Internal Precision: 128
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0001296015753740167:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.2
Target0.3
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 < 0.0001296015753740167

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.0001296015753740167 < x

    1. Initial program 0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0001296015753740167:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(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: 12.4s)Debug log

sample151.0ms

Algorithm
intervals

simplify3.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
3.0ms
(log (+ 1 x))

prune7.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 42.0b

localize12.0ms

Local error

Found 1 expressions with local error:

3.8b
(log (+ 1 x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
3×log-prod
2×add-cube-cbrt
2×add-exp-log
2×*-un-lft-identity
2×log-div
2×pow1
2×add-sqr-sqrt
1×add-log-exp
1×flip-+
1×log-pow
1×rem-log-exp
1×add-cbrt-cube
1×flip3-+
Counts
1 → 14
Calls
1 calls:
Slowest
2.0ms
(log (+ 1 x))

series40.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
40.0ms
(log (+ 1 x))

simplify709.0ms

Counts
9 → 17
Calls
9 calls:
Slowest
380.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
146.0ms
(- (+ x (* 1/3 (pow x 3))) (* 1/2 (pow x 2)))
138.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))
15.0ms
(log 1)
8.0ms
(log (- (* 1 1) (* x x)))

prune155.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize27.0ms

Local error

Found 2 expressions with local error:

0.4b
(* x 1/3)
0.1b
(* (- (* x 1/3) 1/2) (* x x))

rewrite18.0ms

Algorithm
rewrite-expression-head
Rules
4×add-exp-log
4×add-cbrt-cube
4×pow1
3×add-cube-cbrt
3×associate-*l*
3×*-un-lft-identity
3×add-sqr-sqrt
2×add-log-exp
2×*-commutative
2×associate-*l/
1×flip--
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×flip3--
Counts
2 → 25
Calls
2 calls:
Slowest
17.0ms
(* (- (* x 1/3) 1/2) (* x x))
0.0ms
(* x 1/3)

series50.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
37.0ms
(* (- (* x 1/3) 1/2) (* x x))
13.0ms
(* x 1/3)

simplify1.4s

Counts
13 → 31
Calls
13 calls:
Slowest
420.0ms
(* (- (* (* x 1/3) (* x 1/3)) (* 1/2 1/2)) (* x x))
406.0ms
(* (- (pow (* x 1/3) 3) (pow 1/2 3)) (* x x))
344.0ms
(* (* (* (- (* x 1/3) 1/2) (- (* x 1/3) 1/2)) (- (* x 1/3) 1/2)) (* (* (* x x) (* x x)) (* x x)))
55.0ms
(- (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))
55.0ms
(- (* 1/3 (pow x 3)) (* 1/2 (pow x 2)))

prune171.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

3.8b
(log (+ 1 x))
3.8b
(log (+ 1 x))
3.8b
(log (+ 1 x))
0.2b
(* (cbrt (log (+ 1 x))) (cbrt (log (+ 1 x))))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
14×pow1
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×add-exp-log
9×log-prod
8×cbrt-prod
7×associate-*l*
7×associate-*r*
6×log-div
5×log-pow
4×add-log-exp
4×pow1/3
4×add-cbrt-cube
3×flip-+
3×rem-log-exp
3×flip3-+
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-plus
1×pow2
Counts
4 → 72
Calls
4 calls:
Slowest
7.0ms
(* (cbrt (log (+ 1 x))) (cbrt (log (+ 1 x))))
1.0ms
(log (+ 1 x))
1.0ms
(log (+ 1 x))
1.0ms
(log (+ 1 x))

series139.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
73.0ms
(* (cbrt (log (+ 1 x))) (cbrt (log (+ 1 x))))
22.0ms
(log (+ 1 x))
22.0ms
(log (+ 1 x))
22.0ms
(log (+ 1 x))

simplify2.6s

Counts
52 → 84
Calls
52 calls:
Slowest
349.0ms
(- (+ (* 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)))))))
298.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
297.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
265.0ms
(- (+ (* 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)))))
217.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))

prune729.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

3.9b
(log (sqrt (+ 1 x)))
3.9b
(log (sqrt (+ 1 x)))
3.8b
(log (+ 1 x))
3.8b
(log (+ 1 x))

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
18×log-prod
10×add-cube-cbrt
10×*-un-lft-identity
10×add-sqr-sqrt
8×add-exp-log
8×log-div
8×pow1
6×sqrt-prod
6×log-pow
4×add-log-exp
4×flip-+
4×rem-log-exp
4×sqrt-div
4×add-cbrt-cube
4×flip3-+
2×pow1/2
Counts
4 → 64
Calls
4 calls:
Slowest
3.0ms
(log (sqrt (+ 1 x)))
2.0ms
(log (sqrt (+ 1 x)))
2.0ms
(log (+ 1 x))
2.0ms
(log (+ 1 x))

series157.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
54.0ms
(log (sqrt (+ 1 x)))
46.0ms
(log (sqrt (+ 1 x)))
32.0ms
(log (+ 1 x))
25.0ms
(log (+ 1 x))

simplify1.5s

Counts
42 → 76
Calls
42 calls:
Slowest
369.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
295.0ms
(- (+ (/ 1 x) (log -1)) (+ (log (/ -1 x)) (* 1/2 (/ 1 (pow x 2)))))
153.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))
152.0ms
(- (+ (* 1/2 x) (* 1/6 (pow x 3))) (* 1/4 (pow x 2)))
132.0ms
(- (/ 1 x) (+ (log (/ 1 x)) (* 1/2 (/ 1 (pow x 2)))))

prune788.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes109.0ms

Accuracy

99.2% (0.2b remaining)

Error of 0.2b against oracle of 0.0b and baseline of 20.7b

bsearch43.0ms

end0.0ms

sample3.5s

Algorithm
intervals