Average Error: 0.2 → 0.2
Time: 21.5s
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left((\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right) + \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right))_*\right|\]

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. Initial simplification0.2

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

  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. Simplified0.2

    \[\leadsto \color{blue}{\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\left|x\right| \cdot 2\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{6} \cdot \left|x\right|\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right))_*\right|}\]

  5. Final simplification0.2

    \[\leadsto \left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left((\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right) + \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right))_*\right|\]

Reproduce

herbie shell --seed 2018362 +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: 11.1s)Debug log

start249.0ms

Algorithm
intervals

setup479.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.1b

localize67.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 1/21 (sqrt PI))
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.4b
(* 2/3 (fabs x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

1.0ms
(* 2/3 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

series9.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

9.0ms
(* 2/3 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.0s

Counts
12 → 54
Calls

12 calls. Slowest were:

369.0ms
(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))))
343.0ms
(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))))
288.0ms
(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))))

prune1.5s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize53.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(/ 1/21 (sqrt PI))
0.4b
(* 2/3 (fabs x))

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

1.0ms
(* 2/3 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

series9.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

9.0ms
(* 2/3 (fabs x))
0.0ms
(sqrt PI)
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

simplify628.0ms

Counts
12 → 54
Calls

12 calls. Slowest were:

224.0ms
(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))))
202.0ms
(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))))
191.0ms
(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))))

prune1.4s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.1b

localize23.0ms

Local error

Found 4 expressions with local error:

1.0b
(/ 1/21 (sqrt PI))
1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.4b
(* 2/3 (fabs x))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

1.0ms
(* 2/3 (fabs x))
1.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

series11.0ms

Counts
4 → 6
Calls

4 calls. Slowest were:

11.0ms
(* 2/3 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.8s

Counts
12 → 54
Calls

12 calls. Slowest were:

634.0ms
(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)))))
596.0ms
(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)))))
583.0ms
(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)))))

prune1.4s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize41.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
1.0b
(/ 1/21 (sqrt PI))
0.3b
(* (fabs x) 1/5)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

5.0ms
(* (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)

simplify970.0ms

Counts
11 → 54
Calls

11 calls. Slowest were:

336.0ms
(fabs (fma (/ (fabs x) (sqrt PI)) (fma (* (fabs x) (fabs x)) (fma (* (fabs x) 1/5) (fabs x) 2/3) 2) (* (pow (fabs x) 7) (/ 1/21 (sqrt PI)))))
318.0ms
(fabs (fma (/ (fabs x) (sqrt PI)) (fma (* (fabs x) (fabs x)) (fma (* (fabs x) 1/5) (fabs x) 2/3) 2) (* (pow (fabs x) 7) (/ 1/21 (sqrt PI)))))
297.0ms
(fabs (fma (/ (fabs x) (sqrt PI)) (fma (* (fabs x) (fabs x)) (fma (* (fabs x) 1/5) (fabs x) 2/3) 2) (* (pow (fabs x) 7) (/ 1/21 (sqrt PI)))))

prune1.3s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

regimes126.0ms

Accuracy

0% (0.1b remaining)

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

bsearch5.0ms