Average Error: 58.7 → 0.4
Time: 10.1s
Precision: 64
Internal Precision: 128
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}\]

  4. Final simplification0.4

    \[\leadsto x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]

Reproduce

herbie shell --seed 323378853 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))

Details

Time bar (total: 9.4s)Debug log

sample115.0ms

Algorithm
intervals
Results
45.0ms110×body1280valid
22.0ms78×body640valid
11.0ms374×pre80true
7.0ms39×body320valid
6.0ms122×body80valid
3.0ms25×body160valid

simplify7.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
7.0ms
(- (exp x) 1)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 58.6b

localize25.0ms

Local error

Found 1 expressions with local error:

6.1b
(- (exp x) 1)

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
2×add-sqr-sqrt
1×add-log-exp
1×flip--
1×add-cube-cbrt
1×add-exp-log
1×difference-of-sqr-1
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×sub-neg
1×pow1
Counts
1 → 12
Calls
1 calls:
Slowest
8.0ms
(- (exp x) 1)

series23.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
23.0ms
(- (exp x) 1)

simplify176.0ms

Counts
4 → 15
Calls
4 calls:
Slowest
157.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
7.0ms
(- (exp x) 1)
7.0ms
(- (exp x) 1)
4.0ms
(+ (sqrt (exp x)) 1)

prune188.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize35.0ms

Local error

Found 2 expressions with local error:

0.3b
(* x 1/6)
0.0b
(* (+ (* x 1/6) 1/2) (* x x))

rewrite17.0ms

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
16.0ms
(* (+ (* x 1/6) 1/2) (* x x))
0.0ms
(* x 1/6)

series71.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
57.0ms
(* (+ (* x 1/6) 1/2) (* x x))
15.0ms
(* x 1/6)

simplify1.6s

Counts
13 → 31
Calls
13 calls:
Slowest
609.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
529.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
146.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))
68.0ms
(+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))
65.0ms
(+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2)))

prune358.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize21.0ms

Local error

Found 3 expressions with local error:

2.7b
(log (exp (* (+ (* x 1/6) 1/2) (* x x))))
0.3b
(* x 1/6)
0.0b
(* (+ (* x 1/6) 1/2) (* x x))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
6×pow1
5×add-cube-cbrt
5×add-exp-log
5×add-cbrt-cube
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
3×log-pow
3×associate-*l*
3×log-prod
2×*-commutative
2×associate-*l/
1×flip-+
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×associate-*r*
1×rem-log-exp
1×exp-to-pow
1×flip3-+
1×exp-prod
Counts
3 → 39
Calls
3 calls:
Slowest
13.0ms
(* (+ (* x 1/6) 1/2) (* x x))
6.0ms
(log (exp (* (+ (* x 1/6) 1/2) (* x x))))
0.0ms
(* x 1/6)

series130.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
59.0ms
(* (+ (* x 1/6) 1/2) (* x x))
57.0ms
(log (exp (* (+ (* x 1/6) 1/2) (* x x))))
15.0ms
(* x 1/6)

simplify2.0s

Counts
22 → 48
Calls
22 calls:
Slowest
624.0ms
(* (- (* (* x 1/6) (* x 1/6)) (* 1/2 1/2)) (* x x))
522.0ms
(* (+ (pow (* x 1/6) 3) (pow 1/2 3)) (* x x))
148.0ms
(* (* (* (+ (* x 1/6) 1/2) (+ (* x 1/6) 1/2)) (+ (* x 1/6) 1/2)) (* (* (* x x) (* x x)) (* x x)))
113.0ms
(log (* (cbrt (exp (* (+ (* x 1/6) 1/2) (* x x)))) (cbrt (exp (* (+ (* x 1/6) 1/2) (* x x))))))
111.0ms
(log (sqrt (exp (* (+ (* x 1/6) 1/2) (* x x)))))

prune689.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.1b

localize1.0ms

Local error

Found 1 expressions with local error:

6.1b
(- (exp x) 1)

rewrite7.0ms

Algorithm
rewrite-expression-head
Rules
2×add-sqr-sqrt
1×add-log-exp
1×flip--
1×add-cube-cbrt
1×add-exp-log
1×difference-of-sqr-1
1×add-cbrt-cube
1×flip3--
1×*-un-lft-identity
1×sub-neg
1×pow1
Counts
1 → 12
Calls
1 calls:
Slowest
7.0ms
(- (exp x) 1)

series24.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
24.0ms
(- (exp x) 1)

simplify177.0ms

Counts
4 → 15
Calls
4 calls:
Slowest
158.0ms
(+ x (+ (* 1/6 (pow x 3)) (* 1/2 (pow x 2))))
7.0ms
(- (exp x) 1)
7.0ms
(- (exp x) 1)
4.0ms
(+ (sqrt (exp x)) 1)

prune133.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.1b

regimes227.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch3.0ms

end0.0ms

sample3.4s

Algorithm
intervals
Results
1.4s3385×body1280valid
741.0ms2504×body640valid
403.0ms11929×pre80true
240.0ms1292×body320valid
237.0ms4117×body80valid
77.0ms631×body160valid