Average Error: 0.2 → 0.2
Time: 14.6s
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) + \left(2 \cdot \left|x\right|\right))_*\right) + \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{1}{21} \cdot \frac{1}{\sqrt{\pi}}\right)\right) \cdot {\left(\left|x\right|\right)}^{5}\right))_*\right|\]

Error

Bits error versus x

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. Simplified0.2

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

  3. Simplified0.1

    \[\leadsto \color{blue}{\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left((\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) + \left(2 \cdot \left|x\right|\right))_*\right) + \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right) \cdot {\left(\left|x\right|\right)}^{5}\right))_*\right|}\]
  4. Using strategy rm

  5. Applied div-inv0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019002 +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: 13.9s)Debug log

sample232.0ms

Algorithm
intervals

simplify449.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

449.0ms
(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

localize85.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 1/21 (sqrt PI))
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.4b
(* 1/5 (fabs x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

2.0ms
(* 1/5 (fabs x))
1.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

series13.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

13.0ms
(* 1/5 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.4s

Counts
12 → 54
Calls

12 calls. Slowest were:

729.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (* (* 1/5 (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x)) (fma (* 2/3 (fabs x)) (* (fabs x) (fabs x)) (* (fabs x) 2))) (* (* (/ 1/21 (sqrt PI)) (fabs x)) (pow (* (fabs x) (fabs x)) 3))))
342.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (* (* 1/5 (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x)) (fma (* 2/3 (fabs x)) (* (fabs x) (fabs x)) (* (fabs x) 2))) (* (* (/ 1/21 (sqrt PI)) (fabs x)) (pow (* (fabs x) (fabs x)) 3))))
292.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (* (* 1/5 (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x)) (fma (* 2/3 (fabs x)) (* (fabs x) (fabs x)) (* (fabs x) 2))) (* (* (/ 1/21 (sqrt PI)) (fabs x)) (pow (* (fabs x) (fabs x)) 3))))

prune903.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize55.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(/ 1/21 (sqrt PI))
0.4b
(* (fabs x) 1/5)

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

12.0ms
(* (fabs x) 1/5)
1.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

series15.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

15.0ms
(* (fabs x) 1/5)
0.0ms
(sqrt PI)
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

simplify725.0ms

Counts
11 → 54
Calls

11 calls. Slowest were:

281.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (* (fabs x) (fabs x)) (fma (* (fabs x) 1/5) (* (fabs x) (fabs x)) (* 2/3 (fabs x))) (* 2 (fabs x))) (* (* (* (fabs x) (fabs x)) (/ 1/21 (sqrt PI))) (pow (fabs x) 5))))
239.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (* (fabs x) (fabs x)) (fma (* (fabs x) 1/5) (* (fabs x) (fabs x)) (* 2/3 (fabs x))) (* 2 (fabs x))) (* (* (* (fabs x) (fabs x)) (/ 1/21 (sqrt PI))) (pow (fabs x) 5))))
190.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (* (fabs x) (fabs x)) (fma (* (fabs x) 1/5) (* (fabs x) (fabs x)) (* 2/3 (fabs x))) (* 2 (fabs x))) (* (* (* (fabs x) (fabs x)) (/ 1/21 (sqrt PI))) (pow (fabs x) 5))))

prune995.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize28.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 1/21 (sqrt PI))
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.4b
(* 1/5 (fabs x))

rewrite5.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

2.0ms
(* 1/5 (fabs x))
1.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

series13.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

13.0ms
(* 1/5 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.7s

Counts
12 → 54
Calls

12 calls. Slowest were:

624.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (fma (* 1/5 (fabs x)) (* (fabs x) (fabs x)) (* 2/3 (fabs x))) (* (fabs x) (fabs x)) (* (fabs x) 2)) (* (* (pow (fabs x) 6) (fabs x)) (/ 1/21 (sqrt PI)))))
577.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (fma (* 1/5 (fabs x)) (* (fabs x) (fabs x)) (* 2/3 (fabs x))) (* (fabs x) (fabs x)) (* (fabs x) 2)) (* (* (pow (fabs x) 6) (fabs x)) (/ 1/21 (sqrt PI)))))
474.0ms
(fabs (fma (/ 1 (sqrt PI)) (fma (fma (* 1/5 (fabs x)) (* (fabs x) (fabs x)) (* 2/3 (fabs x))) (* (fabs x) (fabs x)) (* (fabs x) 2)) (* (* (pow (fabs x) 6) (fabs x)) (/ 1/21 (sqrt PI)))))

prune1.0s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize34.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(* 1/21 (/ 1 (sqrt PI)))
0.4b
(* (fabs x) 1/5)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 49
Calls

4 calls. Slowest were:

4.0ms
(* (fabs x) 1/5)
2.0ms
(* 1/21 (/ 1 (sqrt PI)))
0.0ms
(sqrt PI)

series9.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

9.0ms
(* (fabs x) 1/5)
0.0ms
(sqrt PI)
0.0ms
(* 1/21 (/ 1 (sqrt PI)))
0.0ms
(sqrt PI)

simplify16.0ms

Counts
11 → 55
Calls

11 calls. Slowest were:

2.0ms
(sqrt (/ 1 (sqrt PI)))
2.0ms
(cbrt (/ 1 (sqrt PI)))
1.0ms
(* 1/5 (fabs x))

prune1.0s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

regimes128.0ms

Accuracy

0% (0.2b remaining)

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

bsearch5.0ms

end0.0ms

sample5.1s

Algorithm
intervals