Average Error: 0.2 → 0.2
Time: 43.6s
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3}\right) + 2 \cdot \left|x\right|\right)\right)\right|\]

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 inf 0.2

    \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)}\right|\]

  3. Final simplification0.2

    \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3}\right) + 2 \cdot \left|x\right|\right)\right)\right|\]

Reproduce

herbie shell --seed 2019018 
(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: 42.9s)Debug log

sample218.0ms

Algorithm
intervals
Results
126.0ms442×body80valid

simplify278.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
278.0ms
(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))))))

prune41.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize70.0ms

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 (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)))))
0.1b
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))

rewrite1.4s

Algorithm
rewrite-expression-head
Rules
37×add-exp-log
35×add-cbrt-cube
35×pow1
23×prod-exp
22×cbrt-unprod
22×pow-prod-down
15×associate-*l/
12×frac-add
12×associate-*l*
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×flip-+
6×associate-*r/
6×flip3-+
6×frac-times
5×associate-*r*
4×add-log-exp
3×div-inv
3×*-commutative
1×pow1/2
1×distribute-lft-in
1×rec-exp
1×distribute-rgt-in
Counts
4 → 94
Calls
4 calls:
Slowest
1.2s
(* (/ 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)))))
184.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
23.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)

series314.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
261.0ms
(* (/ 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)))))
28.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
25.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)

simplify18.6s

Counts
72 → 104
Calls
72 calls:
Slowest
738.0ms
(* (* (* (/ 1 (sqrt PI)) (/ 1 (sqrt PI))) (/ 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)))) (+ (+ (+ (* 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))))) (+ (+ (+ (* 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))))))
735.0ms
(* (/ 1 (sqrt PI)) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))))
693.0ms
(* 1 (+ (* (+ (pow (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) 3) (pow (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))) 3)) 21) (* (+ (* (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))) (- (* (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))) (* 1 (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))))))
667.0ms
(* (+ (* (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))) (- (* (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))) 21)
665.0ms
(* (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))) (/ 1 (sqrt PI)))

prune1.9s

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0b

localize71.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 2/3 (pow (fabs x) 3))
0.2b
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
0.1b
(* 1/5 (pow (fabs x) 5))
0.1b
(* 1/21 (pow (fabs x) 7))

rewrite53.0ms

Algorithm
rewrite-expression-head
Rules
26×associate-*r*
14×add-cube-cbrt
14×*-un-lft-identity
14×add-sqr-sqrt
9×unpow-prod-down
7×associate-*l*
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×sqrt-prod
4×*-commutative
3×cube-prod
3×sqrt-div
2×flip-+
2×associate-*r/
2×flip3-+
2×frac-times
1×div-inv
1×cbrt-unprod
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×unpow3
1×distribute-lft-in
1×cube-mult
1×distribute-rgt-in
Counts
4 → 75
Calls
4 calls:
Slowest
41.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
4.0ms
(* 1/5 (pow (fabs x) 5))
4.0ms
(* 1/21 (pow (fabs x) 7))
2.0ms
(* 2/3 (pow (fabs x) 3))

series358.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
238.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
47.0ms
(* 1/5 (pow (fabs x) 5))
38.0ms
(* 1/21 (pow (fabs x) 7))
35.0ms
(* 2/3 (pow (fabs x) 3))

simplify3.9s

Counts
56 → 87
Calls
56 calls:
Slowest
403.0ms
(* (* (* (sqrt (/ 1 PI)) (sqrt (/ 1 PI))) (sqrt (/ 1 PI))) (* (* (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))))
323.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
299.0ms
(* (sqrt PI) (- (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
298.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
288.0ms
(- (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))

prune1.6s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize41.0ms

Local error

Found 4 expressions with local error:

7.8b
(sqrt (pow (fabs x) 7))
7.8b
(sqrt (pow (fabs x) 7))
0.2b
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7)))))))
0.2b
(* 2/3 (pow (fabs x) 3))

rewrite75.0ms

Algorithm
rewrite-expression-head
Rules
18×add-sqr-sqrt
16×sqrt-prod
14×add-cube-cbrt
14×associate-*r*
14×*-un-lft-identity
11×unpow-prod-down
7×associate-*l*
6×add-exp-log
6×add-cbrt-cube
6×pow1
4×add-log-exp
4×rem-sqrt-square
3×cube-prod
3×sqrt-div
2×flip-+
2×*-commutative
2×associate-*r/
2×pow1/2
2×flip3-+
2×frac-times
1×div-inv
1×cbrt-unprod
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×unpow3
1×distribute-lft-in
1×cube-mult
1×distribute-rgt-in
Counts
4 → 79
Calls
4 calls:
Slowest
67.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7)))))))
2.0ms
(sqrt (pow (fabs x) 7))
2.0ms
(sqrt (pow (fabs x) 7))
2.0ms
(* 2/3 (pow (fabs x) 3))

series394.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
261.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7)))))))
52.0ms
(sqrt (pow (fabs x) 7))
47.0ms
(sqrt (pow (fabs x) 7))
33.0ms
(* 2/3 (pow (fabs x) 3))

simplify4.1s

Counts
60 → 91
Calls
60 calls:
Slowest
459.0ms
(* (* (* (sqrt (/ 1 PI)) (sqrt (/ 1 PI))) (sqrt (/ 1 PI))) (* (* (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7)))))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7))))))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7))))))))
364.0ms
(- (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7))))))
328.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
320.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (* 1/21 (sqrt (pow (fabs x) 7))) (sqrt (pow (fabs x) 7)))))))
320.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))

prune1.8s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize41.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.4b
(* 1/21 (/ (fabs x) (sqrt PI)))
0.3b
(/ (fabs x) (sqrt PI))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
19×add-cube-cbrt
19×*-un-lft-identity
19×add-sqr-sqrt
18×times-frac
13×associate-*r*
6×add-exp-log
6×add-cbrt-cube
4×add-log-exp
4×pow1
3×associate-/l*
3×associate-/r*
2×div-inv
2×pow1/2
1×*-commutative
1×associate-*r/
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 65
Calls
4 calls:
Slowest
3.0ms
(/ (fabs x) (sqrt PI))
2.0ms
(* 1/21 (/ (fabs x) (sqrt PI)))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series255.0ms

Counts
4 → 8
Calls
4 calls:
Slowest
135.0ms
(/ (fabs x) (sqrt PI))
120.0ms
(* 1/21 (/ (fabs x) (sqrt PI)))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify696.0ms

Counts
47 → 73
Calls
47 calls:
Slowest
278.0ms
(/ (* (* (fabs x) (fabs x)) (fabs x)) (* (* (sqrt PI) (sqrt PI)) (sqrt PI)))
137.0ms
(fabs (+ (* (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (/ 1 (sqrt PI))) (* (* (pow (fabs x) 3) (pow (fabs x) 3)) (* 1/21 (/ (fabs x) (sqrt PI))))))
126.0ms
(fabs (+ (* (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (/ 1 (sqrt PI))) (* (* (pow (fabs x) 3) (pow (fabs x) 3)) (* 1/21 (/ (fabs x) (sqrt PI))))))
16.0ms
(/ (* (cbrt (fabs x)) (cbrt (fabs x))) 1)
16.0ms
(/ (* (cbrt (fabs x)) (cbrt (fabs x))) (* (cbrt (sqrt PI)) (cbrt (sqrt PI))))

prune1.3s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

regimes93.0ms

Accuracy

0% (0.2b remaining)

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

bsearch3.0ms

end0.0ms

sample5.3s

Algorithm
intervals
Results
5.0s13821×body80valid