Average Error: 0.2 → 0.2
Time: 36.8s
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \frac{1}{21}\right) \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right) + \left(\left(2 \cdot \left|x\right| + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

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. Using strategy rm

  3. Applied distribute-rgt-in0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019004 
(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: 36.1s)Debug log

sample247.0ms

Algorithm
intervals

simplify254.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

254.0ms
(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))))))

prune22.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize83.0ms

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
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)))))
148.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
25.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))

series359.0ms

Counts
4 → 10
Calls

4 calls. Slowest were:

288.0ms
(* (/ 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)))))
38.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
33.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)

simplify18.6s

Counts
72 → 104
Calls

72 calls. Slowest were:

778.0ms
(* (* (* (/ 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))))))
762.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
748.0ms
(* (/ 1 (sqrt PI)) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))))

prune1.9s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 4 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

3.0ms
(/ (fabs x) (sqrt PI))
2.0ms
(* 1/21 (/ (fabs x) (sqrt PI)))
0.0ms
(sqrt PI)

series224.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

138.0ms
(* 1/21 (/ (fabs x) (sqrt PI)))
86.0ms
(/ (fabs x) (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify674.0ms

Counts
47 → 73
Calls

47 calls. Slowest were:

286.0ms
(/ (* (* (fabs x) (fabs x)) (fabs x)) (* (* (sqrt PI) (sqrt PI)) (sqrt PI)))
148.0ms
(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))))))
113.0ms
(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))))))

prune1.4s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize77.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.3b
(* 1/5 (fabs x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 35
Calls

4 calls. Slowest were:

1.0ms
(* 1/5 (fabs x))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series8.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

8.0ms
(* 1/5 (fabs x))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify976.0ms

Counts
9 → 41
Calls

9 calls. Slowest were:

359.0ms
(fabs (+ (+ (* (* (/ (fabs x) (sqrt PI)) 1/21) (* (fabs x) (pow (fabs x) 5))) (* (* (* 1/5 (fabs x)) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (/ 1 (sqrt PI))))) (* (* (/ 1 (sqrt PI)) (fabs x)) (+ 2 (* 2/3 (* (fabs x) (fabs x)))))))
324.0ms
(fabs (+ (+ (* (* (/ (fabs x) (sqrt PI)) 1/21) (* (fabs x) (pow (fabs x) 5))) (* (* (* 1/5 (fabs x)) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (/ 1 (sqrt PI))))) (* (* (/ 1 (sqrt PI)) (fabs x)) (+ 2 (* 2/3 (* (fabs x) (fabs x)))))))
286.0ms
(fabs (+ (+ (* (* (/ (fabs x) (sqrt PI)) 1/21) (* (fabs x) (pow (fabs x) 5))) (* (* (* 1/5 (fabs x)) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (/ 1 (sqrt PI))))) (* (* (/ 1 (sqrt PI)) (fabs x)) (+ 2 (* 2/3 (* (fabs x) (fabs x)))))))

prune747.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.3b
(* 1/21 (* (fabs x) (/ 1 (sqrt PI))))
0.2b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

45.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
4.0ms
(* 1/21 (* (fabs x) (/ 1 (sqrt PI))))
0.0ms
(sqrt PI)

series172.0ms

Counts
4 → 8
Calls

4 calls. Slowest were:

137.0ms
(* 1/21 (* (fabs x) (/ 1 (sqrt PI))))
35.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.9s

Counts
22 → 58
Calls

22 calls. Slowest were:

563.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
414.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
229.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))

prune1.1s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0b

regimes113.0ms

Accuracy

0% (0.2b remaining)

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

bsearch4.0ms

end0.0ms

sample5.8s

Algorithm
intervals