Average Error: 0.2 → 0.1
Time: 32.5s
Precision: 64
Internal Precision: 128
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) + \left(2 \cdot \left|x\right|\right))_*\right) + \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right) \cdot {\left(\left|x\right|\right)}^{5}\right))_*\right|\]

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019022 +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: 31.7s)Debug log

sample165.0ms

Algorithm
intervals
Results
137.0ms421×body80valid

simplify381.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
381.0ms
(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

localize84.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.1b
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
0.1b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.1b
(* (* (fabs x) (fabs x)) (fabs x))

rewrite215.0ms

Algorithm
rewrite-expression-head
Rules
47×pow1
38×add-exp-log
38×add-cbrt-cube
24×cbrt-unprod
24×prod-exp
24×pow-prod-down
9×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow-prod-up
5×associate-*r*
5×pow-plus
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×*-commutative
2×div-inv
2×associate-*l/
2×pow2
1×pow3
1×pow1/2
Counts
4 → 93
Calls
4 calls:
Slowest
179.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
22.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
12.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

series83.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
34.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
33.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
15.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

simplify8.9s

Counts
62 → 103
Calls
62 calls:
Slowest
1.0s
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
563.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
513.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
502.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
494.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))

prune1.9s

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.1b

localize35.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.1b
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
0.1b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.1b
(* (* (fabs x) (fabs x)) (fabs x))

rewrite315.0ms

Algorithm
rewrite-expression-head
Rules
47×pow1
38×add-exp-log
38×add-cbrt-cube
24×cbrt-unprod
24×prod-exp
24×pow-prod-down
9×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow-prod-up
5×associate-*r*
5×pow-plus
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×*-commutative
2×div-inv
2×associate-*l/
2×pow2
1×pow3
1×pow1/2
Counts
4 → 93
Calls
4 calls:
Slowest
250.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
43.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
19.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

series86.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
47.0ms
(* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))
24.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
14.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

simplify8.4s

Counts
62 → 103
Calls
62 calls:
Slowest
641.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
557.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
524.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
501.0ms
(* (* (* (/ 1 5) (/ 1 5)) (/ 1 5)) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
497.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))

prune2.0s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize29.0ms

Local error

Found 4 expressions with local error:

18.6b
(sqrt (* (* (fabs x) (fabs x)) (fabs x)))
18.6b
(sqrt (* (* (fabs x) (fabs x)) (fabs x)))
1.0b
(sqrt PI)
0.2b
(* (sqrt (* (* (fabs x) (fabs x)) (fabs x))) (sqrt (* (* (fabs x) (fabs x)) (fabs x))))

rewrite33.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
7×pow1/2
6×add-cube-cbrt
6×add-exp-log
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×sqrt-prod
4×associate-*l*
4×associate-*r*
4×expm1-log1p-u
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×pow2
Counts
4 → 60
Calls
4 calls:
Slowest
24.0ms
(* (sqrt (* (* (fabs x) (fabs x)) (fabs x))) (sqrt (* (* (fabs x) (fabs x)) (fabs x))))
4.0ms
(sqrt (* (* (fabs x) (fabs x)) (fabs x)))
4.0ms
(sqrt (* (* (fabs x) (fabs x)) (fabs x)))
0.0ms
(sqrt PI)

series79.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
33.0ms
(sqrt (* (* (fabs x) (fabs x)) (fabs x)))
32.0ms
(sqrt (* (* (fabs x) (fabs x)) (fabs x)))
14.0ms
(* (sqrt (* (* (fabs x) (fabs x)) (fabs x))) (sqrt (* (* (fabs x) (fabs x)) (fabs x))))
0.0ms
(sqrt PI)

simplify235.0ms

Counts
30 → 70
Calls
30 calls:
Slowest
81.0ms
(* (* (* (sqrt (* (* (fabs x) (fabs x)) (fabs x))) (sqrt (* (* (fabs x) (fabs x)) (fabs x)))) (sqrt (* (* (fabs x) (fabs x)) (fabs x)))) (* (* (sqrt (* (* (fabs x) (fabs x)) (fabs x))) (sqrt (* (* (fabs x) (fabs x)) (fabs x)))) (sqrt (* (* (fabs x) (fabs x)) (fabs x)))))
22.0ms
(* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x)))
19.0ms
(* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x)))
8.0ms
(* (cbrt (sqrt (* (* (fabs x) (fabs x)) (fabs x)))) (cbrt (sqrt (* (* (fabs x) (fabs x)) (fabs x)))))
7.0ms
(sqrt (sqrt (* (* (fabs x) (fabs x)) (fabs x))))

prune1.5s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.1b

localize76.0ms

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
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
3×associate-*r*
3×associate-/r*
2×pow1/2
1×div-inv
1×*-commutative
1×frac-2neg
1×clear-num
Counts
4 → 48
Calls
4 calls:
Slowest
2.0ms
(* 1/5 (fabs x))
1.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series14.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
14.0ms
(* 1/5 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify820.0ms

Counts
12 → 54
Calls
12 calls:
Slowest
274.0ms
(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))))
269.0ms
(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))))
261.0ms
(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))))
2.0ms
(* 1/5 (fabs x))
2.0ms
(* 1/5 (fabs x))

prune990.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes101.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.1b and baseline of 0.1b

bsearch5.0ms

end0.0ms

sample5.3s

Algorithm
intervals
Results
5.1s13911×body80valid