Average Error: 0.2 → 0.1
Time: 18.1s
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. 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. Simplified0.1

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

  5. Taylor expanded around -inf 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019010 +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: 17.3s)Debug log

sample257.0ms

Algorithm
intervals

simplify403.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
402.0ms
(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))))))

prune46.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.2b

localize107.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
0.2b
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
0.1b
(* (* (fabs x) (fabs x)) (fabs x))
0.1b
(* (* (fabs x) (fabs x)) (fabs x))

rewrite69.0ms

Algorithm
rewrite-expression-head
Rules
41×pow1
23×add-exp-log
23×add-cbrt-cube
12×cbrt-unprod
12×prod-exp
12×pow-prod-down
10×pow-prod-up
10×pow-plus
7×add-cube-cbrt
7×associate-*r*
7×*-un-lft-identity
7×add-sqr-sqrt
6×associate-*l*
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
4×pow2
3×*-commutative
2×pow3
1×div-inv
1×associate-*l/
1×pow1/2
Counts
4 → 89
Calls
4 calls:
Slowest
28.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
20.0ms
(* (* (fabs x) (fabs x)) (fabs x))
18.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

series63.0ms

Counts
4 → 10
Calls
4 calls:
Slowest
33.0ms
(* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))
16.0ms
(* (* (fabs x) (fabs x)) (fabs x))
14.0ms
(* (* (fabs x) (fabs x)) (fabs x))
0.0ms
(sqrt PI)

simplify2.3s

Counts
63 → 99
Calls
63 calls:
Slowest
472.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
468.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
313.0ms
(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))))))
218.0ms
(* (* (* (/ 2 3) (/ 2 3)) (/ 2 3)) (* (* (* (* (fabs x) (fabs x)) (* (fabs x) (fabs x))) (* (fabs x) (fabs x))) (* (* (fabs x) (fabs x)) (fabs x))))
122.0ms
(+ (log (/ 2 3)) (log (* (* (fabs x) (fabs x)) (fabs x))))

prune2.0s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize62.0ms

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))

rewrite4.0ms

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)

series13.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
13.0ms
(* 1/5 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.3s

Counts
12 → 54
Calls
12 calls:
Slowest
725.0ms
(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))))
294.0ms
(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))))
251.0ms
(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))))
3.0ms
(* 1/5 (fabs x))
3.0ms
(* 1/5 (fabs x))

prune1.0s

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.1b

localize51.0ms

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)

rewrite7.0ms

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-*l*
3×associate-/r*
2×pow1/2
1×div-inv
1×*-commutative
1×frac-2neg
1×clear-num
Counts
4 → 48
Calls
4 calls:
Slowest
5.0ms
(* (fabs x) 1/5)
1.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series11.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
11.0ms
(* (fabs x) 1/5)
0.0ms
(sqrt PI)
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)

simplify564.0ms

Counts
11 → 54
Calls
11 calls:
Slowest
203.0ms
(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))))
189.0ms
(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))))
162.0ms
(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))))
1.0ms
(* 1/5 (fabs x))
1.0ms
(* 1/5 (fabs x))

prune1.0s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.1b

localize47.0ms

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))

rewrite3.0ms

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

series9.0ms

Counts
4 → 6
Calls
4 calls:
Slowest
9.0ms
(* 1/5 (fabs x))
0.0ms
(/ 1/21 (sqrt PI))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.5s

Counts
12 → 54
Calls
12 calls:
Slowest
531.0ms
(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)))))
527.0ms
(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)))))
464.0ms
(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)))))
3.0ms
(* 1/5 (fabs x))
3.0ms
(* 1/5 (fabs x))

prune945.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

regimes307.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch3.0ms

end0.0ms

sample5.1s

Algorithm
intervals