Average Error: 1.5 → 1.2
Time: 58.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}{\left|x\right|} + \left({\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{15}{8}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right)\right) + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|} + \frac{\frac{3}{4}}{\left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.5

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  2. Simplified1.4

    \[\leadsto \color{blue}{\left(\left(\left(\frac{15}{8} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left({\left(\frac{1}{\left|x\right|}\right)}^{3} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) + e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4} + \frac{\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}}{2 \cdot \left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}}\]

  3. Simplified1.3

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \frac{\frac{1}{\sqrt{\pi}}}{\left|x\right|} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{3}{4}}{\left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt1.2

    \[\leadsto \left(\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \left(\frac{\frac{15}{8}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \frac{\frac{1}{\sqrt{\pi}}}{\left|x\right|} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{3}{4}}{\left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.2

    \[\leadsto \left(\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \left(\frac{\frac{15}{8}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}}}{\left|x\right|} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{3}{4}}{\left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.2

    \[\leadsto \left(\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \left(\frac{\frac{15}{8}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \frac{\frac{1}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}}}}{\left|x\right|} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{3}{4}}{\left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

  10. Final simplification1.2

    \[\leadsto \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}}{\left|x\right|} + \left({\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{15}{8}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right)\right) + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right|}}{\left|x\right|} + \frac{\frac{3}{4}}{\left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right)\]

Reproduce

herbie shell --seed 2019021 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  (* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x)))))))

Details

Time bar (total: 57.3s)Debug log

sample3.4s

Algorithm
intervals
Results
1.3s3512×body80valid

simplify485.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
485.0ms
(* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))))

prune31.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 1.3b

localize96.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.9b
(pow (/ 1 (fabs x)) 4)

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
6×add-exp-log
6×pow1
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×add-cbrt-cube
4×unpow-prod-down
3×pow1/2
3×pow-pow
2×pow-exp
1×inv-pow
1×pow-flip
1×div-inv
1×rec-exp
1×pow-to-exp
Counts
4 → 41
Calls
4 calls:
Slowest
3.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series36.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
35.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify506.0ms

Counts
15 → 47
Calls
15 calls:
Slowest
168.0ms
(+ (+ (* (* (* 15/8 (/ 1 (sqrt PI))) (exp (* (fabs x) (fabs x)))) (* (pow (/ 1 (fabs x)) 3) (pow (/ 1 (fabs x)) 4))) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) (/ 1 (fabs x))))) (* (+ (* (* 3/4 (/ 1 (fabs x))) (pow (/ 1 (fabs x)) 4)) (/ (* (/ 1 (fabs x)) (/ 1 (fabs x))) (* 2 (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
157.0ms
(+ (+ (* (* (* 15/8 (/ 1 (sqrt PI))) (exp (* (fabs x) (fabs x)))) (* (pow (/ 1 (fabs x)) 3) (pow (/ 1 (fabs x)) 4))) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) (/ 1 (fabs x))))) (* (+ (* (* 3/4 (/ 1 (fabs x))) (pow (/ 1 (fabs x)) 4)) (/ (* (/ 1 (fabs x)) (/ 1 (fabs x))) (* 2 (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
145.0ms
(+ (+ (* (* (* 15/8 (/ 1 (sqrt PI))) (exp (* (fabs x) (fabs x)))) (* (pow (/ 1 (fabs x)) 3) (pow (/ 1 (fabs x)) 4))) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) (/ 1 (fabs x))))) (* (+ (* (* 3/4 (/ 1 (fabs x))) (pow (/ 1 (fabs x)) 4)) (/ (* (/ 1 (fabs x)) (/ 1 (fabs x))) (* 2 (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
8.0ms
(* (log (/ 1 (fabs x))) 4)
7.0ms
(pow (sqrt (/ 1 (fabs x))) 4)

prune1.1s

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 0.9b

localize71.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.9b
(pow (/ 1 (fabs x)) 4)

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
6×add-exp-log
6×pow1
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
4×add-cbrt-cube
4×unpow-prod-down
3×pow1/2
3×pow-pow
2×pow-exp
1×inv-pow
1×pow-flip
1×div-inv
1×rec-exp
1×pow-to-exp
Counts
4 → 41
Calls
4 calls:
Slowest
3.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series32.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
32.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify343.0ms

Counts
15 → 47
Calls
15 calls:
Slowest
107.0ms
(+ (+ (* (* (/ (/ 1 (fabs x)) (* (fabs x) (fabs x))) (pow (/ 1 (fabs x)) 4)) (* (/ 15/8 (sqrt PI)) (exp (* (fabs x) (fabs x))))) (* (/ (/ 1 (sqrt PI)) (fabs x)) (exp (* (fabs x) (fabs x))))) (* (+ (/ (/ (/ 1/2 (fabs x)) (fabs x)) (fabs x)) (* (pow (/ 1 (fabs x)) 4) (/ 3/4 (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
96.0ms
(+ (+ (* (* (/ (/ 1 (fabs x)) (* (fabs x) (fabs x))) (pow (/ 1 (fabs x)) 4)) (* (/ 15/8 (sqrt PI)) (exp (* (fabs x) (fabs x))))) (* (/ (/ 1 (sqrt PI)) (fabs x)) (exp (* (fabs x) (fabs x))))) (* (+ (/ (/ (/ 1/2 (fabs x)) (fabs x)) (fabs x)) (* (pow (/ 1 (fabs x)) 4) (/ 3/4 (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
83.0ms
(+ (+ (* (* (/ (/ 1 (fabs x)) (* (fabs x) (fabs x))) (pow (/ 1 (fabs x)) 4)) (* (/ 15/8 (sqrt PI)) (exp (* (fabs x) (fabs x))))) (* (/ (/ 1 (sqrt PI)) (fabs x)) (exp (* (fabs x) (fabs x))))) (* (+ (/ (/ (/ 1/2 (fabs x)) (fabs x)) (fabs x)) (* (pow (/ 1 (fabs x)) 4) (/ 3/4 (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
16.0ms
(* (log (/ 1 (fabs x))) 4)
12.0ms
(* (- (log (fabs x))) 4)

prune968.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.7b

localize24.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×pow1/2
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
Counts
4 → 32
Calls
4 calls:
Slowest
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series0.0ms

Counts
4 → 4
Calls
4 calls:
Slowest
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify2.0ms

Counts
0 → 36
Calls
0 calls:
Slowest

prune813.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.6b

localize38.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)

rewrite5.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×add-cube-cbrt
4×add-exp-log
4×pow1/2
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×add-sqr-sqrt
Counts
4 → 32
Calls
4 calls:
Slowest
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series0.0ms

Counts
4 → 4
Calls
4 calls:
Slowest
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify2.0ms

Counts
0 → 36
Calls
0 calls:
Slowest

prune919.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.6b

regimes338.0ms

Accuracy

0% (0.7b remaining)

Error of 1.2b against oracle of 0.6b and baseline of 1.2b

bsearch5.0ms

end0.0ms

sample48.1s

Algorithm
intervals
Results
43.7s109071×body80valid