Average Error: 1.5 → 0.7
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt[3]{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}} \cdot \left(\sqrt[3]{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}} \cdot \sqrt[3]{\sqrt{\pi}}\right)} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}\right) + \left(\left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot {\left(\left|x\right|\right)}^{-4}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right) + \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 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. Initial simplification1.1

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

  4. Applied inv-pow1.1

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

  5. Applied pow-pow0.6

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

  7. Applied add-cube-cbrt0.6

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

  9. Applied add-cube-cbrt0.7

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

  11. Applied add-cube-cbrt0.7

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

  12. Final simplification0.7

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

Reproduce

herbie shell --seed 2018360 
(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: 12.6s)Debug log

start3.3s

Algorithm
intervals

setup374.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.9b

localize104.0ms

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)

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 41
Calls

4 calls. Slowest were:

2.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series20.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

20.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify537.0ms

Counts
15 → 47
Calls

15 calls. Slowest were:

161.0ms
(+ (+ (/ (* (* 3/4 (/ 1 (fabs x))) (pow (/ 1 (fabs x)) 4)) (/ (sqrt PI) (exp (* (fabs x) (fabs x))))) (* (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8)) (* (/ 1 (pow (fabs x) 3)) (pow (/ 1 (fabs x)) 4)))) (* (+ (/ 1 (fabs x)) (* (* (/ 1 (fabs x)) 1/2) (* (/ 1 (fabs x)) (/ 1 (fabs x))))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
159.0ms
(+ (+ (/ (* (* 3/4 (/ 1 (fabs x))) (pow (/ 1 (fabs x)) 4)) (/ (sqrt PI) (exp (* (fabs x) (fabs x))))) (* (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8)) (* (/ 1 (pow (fabs x) 3)) (pow (/ 1 (fabs x)) 4)))) (* (+ (/ 1 (fabs x)) (* (* (/ 1 (fabs x)) 1/2) (* (/ 1 (fabs x)) (/ 1 (fabs x))))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
149.0ms
(+ (+ (/ (* (* 3/4 (/ 1 (fabs x))) (pow (/ 1 (fabs x)) 4)) (/ (sqrt PI) (exp (* (fabs x) (fabs x))))) (* (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8)) (* (/ 1 (pow (fabs x) 3)) (pow (/ 1 (fabs x)) 4)))) (* (+ (/ 1 (fabs x)) (* (* (/ 1 (fabs x)) 1/2) (* (/ 1 (fabs x)) (/ 1 (fabs x))))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))

prune1.8s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize39.0ms

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)

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 41
Calls

4 calls. Slowest were:

2.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series20.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

20.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify65.0ms

Counts
15 → 47
Calls

15 calls. Slowest were:

17.0ms
(* (log (/ 1 (fabs x))) 4)
8.0ms
(* (- (log (fabs x))) 4)
5.0ms
(pow (* (cbrt (/ 1 (fabs x))) (cbrt (/ 1 (fabs x)))) 4)

prune2.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize22.0ms

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
Counts
4 → 32
Calls

4 calls. Slowest were:

0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series0.0ms

Counts
4 → 4
Calls

4 calls. Slowest were:

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 were:

prune1.7s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize23.0ms

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
Counts
4 → 32
Calls

4 calls. Slowest were:

0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series0.0ms

Counts
4 → 4
Calls

4 calls. Slowest were:

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 were:

prune1.7s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

regimes745.0ms

Accuracy

0% (0.4b remaining)

Error of 0.7b against oracle of 0.3b and baseline of 0.7b

bsearch13.0ms