Average Error: 1.5 → 1.0
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\frac{1}{\left|x\right| \cdot \sqrt{\pi}} + \frac{{\left(\left|x\right|\right)}^{-5}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{\frac{15}{8}}{\sqrt{\pi}}}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{3}{4}}{\left|x\right|}\right)\]

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. 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. Simplified1.2

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\pi} \cdot \left|x\right|} + \frac{{\left(\frac{1}{\left|x\right|}\right)}^{5}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{\frac{15}{8}}{\sqrt{\pi}}}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{\left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}}\]
  5. Using strategy rm

  6. Applied inv-pow1.2

    \[\leadsto \left(\frac{1}{\sqrt{\pi} \cdot \left|x\right|} + \frac{{\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{\frac{15}{8}}{\sqrt{\pi}}}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{\left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

  7. Applied pow-pow1.0

    \[\leadsto \left(\frac{1}{\sqrt{\pi} \cdot \left|x\right|} + \frac{\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{\frac{15}{8}}{\sqrt{\pi}}}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{\left|x\right|} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\]

  8. Final simplification1.0

    \[\leadsto \left(\frac{1}{\left|x\right| \cdot \sqrt{\pi}} + \frac{{\left(\left|x\right|\right)}^{-5}}{\frac{\left|x\right| \cdot \left|x\right|}{\frac{\frac{15}{8}}{\sqrt{\pi}}}}\right) \cdot e^{\left|x\right| \cdot \left|x\right|} + \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{4} \cdot \frac{\frac{3}{4}}{\left|x\right|}\right)\]

Reproduce

herbie shell --seed 2019010 
(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: 59.7s)Debug log

sample3.5s

Algorithm
intervals

simplify423.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
423.0ms
(* (* (/ 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))))))

prune49.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 1.2b

localize148.0ms

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
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
1.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series21.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
21.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify621.0ms

Counts
15 → 47
Calls
15 calls:
Slowest
246.0ms
(+ (+ (* (* (* 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))))
190.0ms
(+ (+ (* (* (* 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))))
155.0ms
(+ (+ (* (* (* 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)
6.0ms
(* (- (log (fabs x))) 4)

prune1.2s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.7b

localize46.0ms

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
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
1.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series26.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
26.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify338.0ms

Counts
15 → 47
Calls
15 calls:
Slowest
102.0ms
(+ (+ (* (* (/ (/ 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))))
97.0ms
(+ (+ (* (* (/ (/ 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))))
94.0ms
(+ (+ (* (* (/ (/ 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))))
13.0ms
(* (log (/ 1 (fabs x))) 4)
11.0ms
(* (- (log (fabs x))) 4)

prune1.1s

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.5b

localize25.0ms

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

prune838.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.4b

localize47.0ms

Local error

Found 4 expressions with local error:

1.1b
(pow (/ 1 (fabs x)) 5)
1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(sqrt PI)

rewrite2.0ms

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

series29.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
29.0ms
(pow (/ 1 (fabs x)) 5)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify539.0ms

Counts
15 → 47
Calls
15 calls:
Slowest
187.0ms
(+ (* (+ (/ 1 (* (sqrt PI) (fabs x))) (/ (pow (/ 1 (fabs x)) 5) (/ (* (fabs x) (fabs x)) (/ 15/8 (sqrt PI))))) (exp (* (fabs x) (fabs x)))) (* (+ (* (/ 3/4 (fabs x)) (pow (/ 1 (fabs x)) 4)) (/ (/ 1/2 (fabs x)) (* (fabs x) (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
181.0ms
(+ (* (+ (/ 1 (* (sqrt PI) (fabs x))) (/ (pow (/ 1 (fabs x)) 5) (/ (* (fabs x) (fabs x)) (/ 15/8 (sqrt PI))))) (exp (* (fabs x) (fabs x)))) (* (+ (* (/ 3/4 (fabs x)) (pow (/ 1 (fabs x)) 4)) (/ (/ 1/2 (fabs x)) (* (fabs x) (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
115.0ms
(+ (* (+ (/ 1 (* (sqrt PI) (fabs x))) (/ (pow (/ 1 (fabs x)) 5) (/ (* (fabs x) (fabs x)) (/ 15/8 (sqrt PI))))) (exp (* (fabs x) (fabs x)))) (* (+ (* (/ 3/4 (fabs x)) (pow (/ 1 (fabs x)) 4)) (/ (/ 1/2 (fabs x)) (* (fabs x) (fabs x)))) (/ (exp (* (fabs x) (fabs x))) (sqrt PI))))
14.0ms
(* (log (/ 1 (fabs x))) 5)
6.0ms
(* (- (log (fabs x))) 5)

prune825.0ms

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 0.2b

regimes481.0ms

Accuracy

0% (0.7b remaining)

Error of 1.0b against oracle of 0.3b and baseline of 1.0b

bsearch4.0ms

end0.0ms

sample49.4s

Algorithm
intervals