Average Error: 0.2 → 0.2
Time: 1.2m
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 2018362 
(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: 1.0m)Debug log

start352.0ms

Algorithm
intervals

setup314.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize73.0ms

Local error

Found 4 expressions with local error:

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

rewrite1.4s

Algorithm
rewrite-expression-head
Counts
4 → 94
Calls

4 calls. Slowest were:

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)))))
166.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
38.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))

series311.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

242.0ms
(* (/ 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)))))
36.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
33.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)

simplify20.7s

Counts
72 → 104
Calls

72 calls. Slowest were:

1.2s
(* (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))) (/ 1 (sqrt PI)))
796.0ms
(* (* (* (/ 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))))
783.0ms
(* (* (* (/ 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))))))

prune2.8s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize51.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 2/3 (pow (fabs x) 3))
0.1b
(* 1/5 (pow (fabs x) 5))
0.1b
(* (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
(* 1/21 (pow (fabs x) 7))

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 75
Calls

4 calls. Slowest were:

49.0ms
(* (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
(* 1/21 (pow (fabs x) 7))
2.0ms
(* 2/3 (pow (fabs x) 3))

series325.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

201.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
45.0ms
(* 1/5 (pow (fabs x) 5))
44.0ms
(* 1/21 (pow (fabs x) 7))
35.0ms
(* 2/3 (pow (fabs x) 3))

simplify4.1s

Counts
56 → 87
Calls

56 calls. Slowest were:

418.0ms
(* (* (* (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)))))))
387.0ms
(- (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))
349.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))

prune2.6s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sqrt (sqrt (/ 1 PI))) (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))))
0.2b
(* (sqrt (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
(* 2/3 (pow (fabs x) 3))
0.1b
(* 1/5 (pow (fabs x) 5))

rewrite114.0ms

Algorithm
rewrite-expression-head
Counts
4 → 104
Calls

4 calls. Slowest were:

69.0ms
(* (sqrt (sqrt (/ 1 PI))) (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))))
39.0ms
(* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
2.0ms
(* 1/5 (pow (fabs x) 5))

series893.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

525.0ms
(* (sqrt (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
(* (sqrt (sqrt (/ 1 PI))) (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))))
46.0ms
(* 1/5 (pow (fabs x) 5))
34.0ms
(* 2/3 (pow (fabs x) 3))

simplify12.0s

Counts
89 → 116
Calls

89 calls. Slowest were:

609.0ms
(* (* (* (sqrt (sqrt (/ 1 PI))) (sqrt (sqrt (/ 1 PI)))) (sqrt (sqrt (/ 1 PI)))) (* (* (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))) (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))) (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))))
571.0ms
(* (* (* (sqrt (sqrt (/ 1 PI))) (sqrt (sqrt (/ 1 PI)))) (sqrt (sqrt (/ 1 PI)))) (* (* (* (sqrt (sqrt (/ 1 PI))) (sqrt (sqrt (/ 1 PI)))) (sqrt (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))))))))
397.0ms
(* (sqrt (sqrt 1)) (* (sqrt (sqrt (/ 1 PI))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))))

prune4.0s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.4b
(* 2/3 (fabs x))
0.3b
(/ (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x))))) (sqrt PI))
0.2b
(* 1/21 (* (fabs x) (fabs x)))

rewrite74.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

68.0ms
(/ (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x))))) (sqrt PI))
4.0ms
(* 1/21 (* (fabs x) (fabs x)))
1.0ms
(* 2/3 (fabs x))

series313.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

288.0ms
(/ (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x))))) (sqrt PI))
15.0ms
(* 1/21 (* (fabs x) (fabs x)))
11.0ms
(* 2/3 (fabs x))
0.0ms
(sqrt PI)

simplify9.8s

Counts
45 → 71
Calls

45 calls. Slowest were:

698.0ms
(/ (sqrt (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x)))))) (* (cbrt (sqrt PI)) (cbrt (sqrt PI))))
612.0ms
(/ (* (cbrt (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x)))))) (cbrt (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x))))))) (sqrt (sqrt PI)))
547.0ms
(/ (sqrt (+ (* (* (fabs x) (pow (fabs x) 4)) (+ 1/5 (* 1/21 (* (fabs x) (fabs x))))) (+ (* 2 (fabs x)) (* (* 2/3 (fabs x)) (* (fabs x) (fabs x)))))) (sqrt PI))

prune1.8s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes127.0ms

Accuracy

0% (0.2b remaining)

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

bsearch6.0ms