Average Error: 0.2 → 0.2
Time: 35.5s
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\color{blue}{{\left(\left|x\right|\right)}^{3}} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  3. Final simplification0.2

    \[\leadsto \left|\left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))

Details

Time bar (total: 34.7s)Debug log

sample285.0ms

Algorithm
intervals

simplify274.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

274.0ms
(fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))))))

prune30.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize74.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.2b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.2b
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
0.1b
(* (* (fabs x) (fabs x)) (fabs x))

rewrite188.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

157.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
21.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
9.0ms
(* (* (fabs x) (fabs x)) (fabs x))

series97.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

47.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
36.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
14.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

simplify8.2s

Counts
62 → 95
Calls

62 calls. Slowest were:

608.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
595.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
548.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))

prune1.9s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.2b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.2b
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
0.1b
(* (* (fabs x) (fabs x)) (fabs x))

rewrite233.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

198.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
22.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
11.0ms
(* (* (fabs x) (fabs x)) (fabs x))

series106.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

48.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
36.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
22.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

simplify8.0s

Counts
62 → 95
Calls

62 calls. Slowest were:

615.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
537.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
500.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))

prune2.0s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize32.0ms

Local error

Found 4 expressions with local error:

33.3b
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
33.3b
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
33.3b
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
1.0b
(sqrt PI)

rewrite52.0ms

Algorithm
rewrite-expression-head
Counts
4 → 38
Calls

4 calls. Slowest were:

19.0ms
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
17.0ms
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
15.0ms
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))

series153.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

61.0ms
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
46.0ms
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
46.0ms
(cbrt (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
0.0ms
(sqrt PI)

simplify89.0ms

Counts
15 → 48
Calls

15 calls. Slowest were:

18.0ms
(cbrt (* 2 (* (* (fabs x) (fabs x)) (fabs x))))
9.0ms
(cbrt (* 2 (* (* (fabs x) (fabs x)) (fabs x))))
8.0ms
(cbrt (* 2 (* (* (fabs x) (fabs x)) (fabs x))))

prune1.0s

Pruning

6 alts after pruning (4 fresh and 2 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.2b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.2b
(* (* (* (fabs x) (fabs x)) (sqrt (fabs x))) (sqrt (fabs x)))
0.2b
(* (* (fabs x) (fabs x)) (sqrt (fabs x)))

rewrite121.0ms

Algorithm
rewrite-expression-head
Counts
4 → 82
Calls

4 calls. Slowest were:

49.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
48.0ms
(* (* (* (fabs x) (fabs x)) (sqrt (fabs x))) (sqrt (fabs x)))
20.0ms
(* (* (fabs x) (fabs x)) (sqrt (fabs x)))

series114.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

52.0ms
(* (* (fabs x) (fabs x)) (sqrt (fabs x)))
35.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
27.0ms
(* (* (* (fabs x) (fabs x)) (sqrt (fabs x))) (sqrt (fabs x)))
0.0ms
(sqrt PI)

simplify3.6s

Counts
64 → 92
Calls

64 calls. Slowest were:

490.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
473.0ms
(* (* (* (* (* (fabs x) (fabs x)) (sqrt (fabs x))) (* (* (fabs x) (fabs x)) (sqrt (fabs x)))) (* (* (fabs x) (fabs x)) (sqrt (fabs x)))) (* (* (sqrt (fabs x)) (sqrt (fabs x))) (sqrt (fabs x))))
441.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))

prune2.5s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

regimes170.0ms

Accuracy

0% (0.2b remaining)

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

bsearch5.0ms

end0.0ms

sample5.4s

Algorithm
intervals