Average Error: 0.2 → 0.2
Time: 54.9s
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 0 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 2019008 +o rules:numerics
(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: 54.2s)Debug log

sample271.0ms

Algorithm
intervals

simplify695.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

695.0ms
(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))))))

prune43.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize80.0ms

Local error

Found 4 expressions with local error:

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

rewrite1.6s

Algorithm
rewrite-expression-head
Counts
4 → 102
Calls

4 calls. Slowest were:

1.4s
(* (/ 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)))))
200.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
37.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))

series327.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

249.0ms
(* (/ 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)))))
49.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
29.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)

simplify21.8s

Counts
72 → 112
Calls

72 calls. Slowest were:

1.0s
(* (* (* (/ 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))))))
947.0ms
(* (* (* (/ 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))))
857.0ms
(* (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)))) (+ (+ (* 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))) (* (/ 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)))))))

prune2.3s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize40.0ms

Local error

Found 4 expressions with local error:

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

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

46.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
5.0ms
(* 2/3 (pow (fabs x) 3))
4.0ms
(* 1/5 (pow (fabs x) 5))

series363.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

243.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
44.0ms
(* 1/5 (pow (fabs x) 5))
42.0ms
(* 1/21 (pow (fabs x) 7))
34.0ms
(* 2/3 (pow (fabs x) 3))

simplify4.1s

Counts
56 → 95
Calls

56 calls. Slowest were:

523.0ms
(* (* (* (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)))))))
471.0ms
(* (sqrt PI) (- (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
347.0ms
(* (sqrt (/ 1 PI)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7)))))

prune1.6s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0b

localize19.0ms

Local error

Found 4 expressions with local error:

7.8b
(sqrt (* 1/21 (pow (fabs x) 7)))
7.8b
(sqrt (* 1/21 (pow (fabs x) 7)))
0.2b
(* 2/3 (pow (fabs x) 3))
0.1b
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7))))))))

rewrite78.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

72.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7))))))))
2.0ms
(* 2/3 (pow (fabs x) 3))
1.0ms
(sqrt (* 1/21 (pow (fabs x) 7)))

series524.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

326.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7))))))))
90.0ms
(sqrt (* 1/21 (pow (fabs x) 7)))
88.0ms
(sqrt (* 1/21 (pow (fabs x) 7)))
21.0ms
(* 2/3 (pow (fabs x) 3))

simplify3.8s

Counts
46 → 85
Calls

46 calls. Slowest were:

548.0ms
(* (* (* (sqrt (/ 1 PI)) (sqrt (/ 1 PI))) (sqrt (/ 1 PI))) (* (* (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7))))))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7)))))))) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7)))))))))
333.0ms
(- (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7)))))))
292.0ms
(* (sqrt 1) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* (sqrt (* 1/21 (pow (fabs x) 7))) (sqrt (* 1/21 (pow (fabs x) 7))))))))

prune1.7s

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0b

localize39.0ms

Local error

Found 4 expressions with local error:

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

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

6.0ms
(/ (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x))))) (sqrt PI))
5.0ms
(* (fabs x) 1/21)
2.0ms
(* 1/5 (fabs x))

series229.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

212.0ms
(/ (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x))))) (sqrt PI))
9.0ms
(* (fabs x) 1/21)
8.0ms
(* 1/5 (fabs x))
0.0ms
(sqrt PI)

simplify7.5s

Counts
40 → 75
Calls

40 calls. Slowest were:

564.0ms
(* (sqrt (/ 1 PI)) (+ (* 1/5 (pow (fabs x) 5)) (+ (* 2 (fabs x)) (+ (* 2/3 (pow (fabs x) 3)) (* 1/21 (pow (fabs x) 7))))))
507.0ms
(/ (* (* (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x))))) (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x)))))) (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x)))))) (* (* (sqrt PI) (sqrt PI)) (sqrt PI)))
382.0ms
(/ (* (cbrt (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x)))))) (cbrt (fma (fabs x) (fma 2/3 (* (fabs x) (fabs x)) 2) (fma (* (* (fabs x) 1/21) (pow (fabs x) 4)) (* (fabs x) (fabs x)) (* (pow (fabs x) 4) (* 1/5 (fabs x))))))) 1)

prune1.1s

Pruning

4 alts after pruning (0 fresh and 4 done)

Merged error: 0b

regimes393.0ms

Accuracy

0% (0.1b remaining)

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

bsearch6.0ms

end0.0ms

sample5.4s

Algorithm
intervals