Average Error: 1.5 → 0.7

Time: 1.1m

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x`

Derivation

Initial program 1.5
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
Simplified1.1
\[\leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left((\left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*}\]
Using strategy rm
Applied inv-pow1.1
\[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left((\left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{4}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]
Applied pow-pow0.7
\[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left((\left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 4\right)}}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]
Final simplification0.7
\[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left((\left(\frac{1}{\left|x\right|} \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right) \cdot \left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot {\left(\left|x\right|\right)}^{-4}\right)\right))_*\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(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

sample3.0s

Algorithm

intervals

Results

3447×	(pre true 80)
3447×	(body real 80)

simplify650.0ms

Counts

1 → 1

Calls

1 calls:

Slowest

650.0ms

(* (* (/ 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))))))

prune28.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.9b

localize97.0ms

Local error

Found 4 expressions with local error:

1.0b	(sqrt PI)
1.0b	(sqrt PI)
1.0b	(pow (/ 1 (fabs x)) 4)
0.5b	(* (/ 1 (fabs x)) (/ 1 (fabs x)))

rewrite11.0ms

Algorithm

rewrite-expression-head

Rules

17×	`pow1`
14×	`add-exp-log`
7×	`inv-pow`
7×	`pow-flip`
7×	`add-cube-cbrt`
7×	`*-un-lft-identity`
7×	`add-sqr-sqrt`
6×	`add-cbrt-cube`
5×	`pow-prod-up`
5×	`rec-exp`
4×	`add-log-exp`
4×	`log1p-expm1-u`
4×	`prod-exp`
4×	`associate-l`
4×	`associate-r`
4×	`unpow-prod-down`
4×	`expm1-log1p-u`
3×	`div-inv`
3×	`pow-prod-down`
3×	`pow-pow`
2×	`pow-exp`
2×	`pow1/2`
1×	`cbrt-unprod`
1×	`*-commutative`
1×	`un-div-inv`
1×	`associate-*r/`
1×	`associate-*l/`
1×	`pow-plus`
1×	`pow-to-exp`
1×	`frac-times`
1×	`pow2`

Counts

4 → 76

Calls

4 calls:

Slowest

8.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
2.0ms	(pow (/ 1 (fabs x)) 4)
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)

series66.0ms

Counts

4 → 8

Calls

4 calls:

Slowest

45.0ms	(pow (/ 1 (fabs x)) 4)
20.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)

simplify667.0ms

Counts

48 → 84

Calls

48 calls:

Slowest

155.0ms	(fma (/ (exp (* (fabs x) (fabs x))) (sqrt PI)) (fma (* (* 3/4 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x)))) (* (/ 1 (fabs x)) (/ 1 (fabs x))) (fma (* 1/2 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x)))) (* (* (/ 1 (pow (fabs x) 3)) (pow (/ 1 (fabs x)) 4)) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8))))
153.0ms	(fma (/ (exp (* (fabs x) (fabs x))) (sqrt PI)) (fma (* (* 3/4 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x)))) (* (/ 1 (fabs x)) (/ 1 (fabs x))) (fma (* 1/2 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x)))) (* (* (/ 1 (pow (fabs x) 3)) (pow (/ 1 (fabs x)) 4)) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8))))
143.0ms	(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
42.0ms	(+ (- (log (fabs x))) (log (/ 1 (fabs x))))
38.0ms	(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))

prune2.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize23.0ms

Local error

Found 4 expressions with local error:

1.0b	(sqrt PI)
1.0b	(sqrt PI)
0.5b	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.5b	(* (/ 1 (fabs x)) (/ 1 (fabs x)))

rewrite17.0ms

Algorithm

rewrite-expression-head

Rules

26×	`pow1`
20×	`add-exp-log`
12×	`inv-pow`
12×	`pow-flip`
10×	`pow-prod-up`
8×	`add-cube-cbrt`
8×	`prod-exp`
8×	`associate-l`
8×	`associate-r`
8×	`add-cbrt-cube`
8×	`*-un-lft-identity`
8×	`rec-exp`
8×	`add-sqr-sqrt`
6×	`pow-prod-down`
4×	`add-log-exp`
4×	`div-inv`
4×	`log1p-expm1-u`
4×	`expm1-log1p-u`
2×	`cbrt-unprod`
2×	`*-commutative`
2×	`un-div-inv`
2×	`associate-*r/`
2×	`associate-*l/`
2×	`pow1/2`
2×	`pow-plus`
2×	`frac-times`
2×	`pow2`

Counts

4 → 94

Calls

4 calls:

Slowest

9.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
6.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)

series48.0ms

Counts

4 → 8

Calls

4 calls:

Slowest

24.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
24.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)

simplify706.0ms

Counts

70 → 102

Calls

70 calls:

Slowest

196.0ms	(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
189.0ms	(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
38.0ms	(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))
26.0ms	(+ (- (log (fabs x))) (log (/ 1 (fabs x))))
26.0ms	(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))

prune2.5s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.2b

localize33.0ms

Local error

Found 4 expressions with local error:

1.0b	(sqrt PI)
1.0b	(sqrt PI)
0.5b	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.5b	(* (/ 1 (fabs x)) (/ 1 (fabs x)))

rewrite12.0ms

Algorithm

rewrite-expression-head

Rules

26×	`pow1`
20×	`add-exp-log`
12×	`inv-pow`
12×	`pow-flip`
10×	`pow-prod-up`
8×	`add-cube-cbrt`
8×	`prod-exp`
8×	`associate-l`
8×	`associate-r`
8×	`add-cbrt-cube`
8×	`*-un-lft-identity`
8×	`rec-exp`
8×	`add-sqr-sqrt`
6×	`pow-prod-down`
4×	`add-log-exp`
4×	`div-inv`
4×	`log1p-expm1-u`
4×	`expm1-log1p-u`
2×	`cbrt-unprod`
2×	`*-commutative`
2×	`un-div-inv`
2×	`associate-*r/`
2×	`associate-*l/`
2×	`pow1/2`
2×	`pow-plus`
2×	`frac-times`
2×	`pow2`

Counts

4 → 94

Calls

4 calls:

Slowest

5.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
5.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)

series47.0ms

Counts

4 → 8

Calls

4 calls:

Slowest

24.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
22.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)

simplify605.0ms

Counts

70 → 102

Calls

70 calls:

Slowest

147.0ms	(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
144.0ms	(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
36.0ms	(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))
36.0ms	(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))
35.0ms	(+ (- (log (fabs x))) (log (/ 1 (fabs x))))

prune2.5s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0.2b

localize57.0ms

Local error

Found 4 expressions with local error:

1.0b	(sqrt PI)
1.0b	(sqrt PI)
1.0b	(expm1 (log1p (sqrt PI)))
0.5b	(* (/ 1 (fabs x)) (/ 1 (fabs x)))

rewrite12.0ms

Algorithm

rewrite-expression-head

Rules

15×	`pow1`
12×	`add-exp-log`
6×	`inv-pow`
6×	`pow-flip`
6×	`add-cube-cbrt`
6×	`add-cbrt-cube`
6×	`*-un-lft-identity`
6×	`add-sqr-sqrt`
5×	`pow-prod-up`
4×	`add-log-exp`
4×	`log1p-expm1-u`
4×	`prod-exp`
4×	`associate-l`
4×	`associate-r`
4×	`rec-exp`
4×	`expm1-log1p-u`
3×	`pow-prod-down`
2×	`div-inv`
2×	`pow1/2`
1×	`expm1-udef`
1×	`cbrt-unprod`
1×	`*-commutative`
1×	`un-div-inv`
1×	`associate-*r/`
1×	`associate-*l/`
1×	`pow-plus`
1×	`expm1-log1p`
1×	`frac-times`
1×	`pow2`

Counts

4 → 68

Calls

4 calls:

Slowest

10.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(sqrt PI)
0.0ms	(expm1 (log1p (sqrt PI)))

series14.0ms

Counts

4 → 6

Calls

4 calls:

Slowest

14.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms	(sqrt PI)
0.0ms	(expm1 (log1p (sqrt PI)))
0.0ms	(sqrt PI)

simplify339.0ms

Counts

34 → 74

Calls

34 calls:

Slowest

159.0ms	(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
32.0ms	(+ (- (log (fabs x))) (log (/ 1 (fabs x))))
30.0ms	(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))
21.0ms	(* (/ 1 (fabs x)) (/ 1 (fabs x)))
8.0ms	(/ 1 (pow (fabs x) 2))

prune1.9s

Pruning

13 alts after pruning (10 fresh and 3 done)

Merged error: 0.2b

regimes573.0ms

Accuracy

0% (0.5b remaining)

Error of 0.7b against oracle of 0.2b and baseline of 0.7b

bsearch8.0ms

end0.0ms

sample50.3s

Algorithm

intervals

Results

108323×	(pre true 80)
108323×	(body real 80)