Average Error: 1.5 → 0.5
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\frac{3}{4} \cdot \left({\left(\left|x\right|\right)}^{-2} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left((\frac{1}{2} \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{15}{8} \cdot \frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{7}}\right)\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 1.5

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

  2. Simplified1.1

    \[\leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left((\left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{4}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*}\]
  3. Using strategy rm

  4. Applied inv-pow1.1

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

  5. Applied pow-pow0.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{3}{4} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left((\left(\frac{1}{2} \cdot \frac{1}{\left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(\frac{1}{{\left(\left|x\right|\right)}^{3}} \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 4\right)}}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]
  6. Using strategy rm

  7. Applied pow10.6

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

  8. Applied pow-flip0.6

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

  9. Applied inv-pow0.6

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

  10. Applied pow-prod-up0.6

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

  11. Simplified0.6

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

  12. Simplified1.0

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

  13. Taylor expanded around 0 0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\frac{3}{4} \cdot \left({\left(\left|x\right|\right)}^{-2} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left((\frac{1}{2} \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\color{blue}{\left(\frac{15}{8} \cdot \left(\frac{1}{{\left(\left|x\right|\right)}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \cdot e^{\left|x\right| \cdot \left|x\right|}\right))_*\]

  14. Simplified0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\frac{3}{4} \cdot \left({\left(\left|x\right|\right)}^{-2} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left((\frac{1}{2} \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\color{blue}{\left(\frac{15}{8} \cdot \frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{7}}\right)} \cdot e^{\left|x\right| \cdot \left|x\right|}\right))_*\]

  15. Final simplification0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\frac{3}{4} \cdot \left({\left(\left|x\right|\right)}^{-2} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left((\frac{1}{2} \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{15}{8} \cdot \frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^{7}}\right)\right))_*\]

Reproduce

herbie shell --seed 2019010 +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

Time bar (total: 1.2m)Debug log

sample3.6s

Algorithm
intervals

simplify644.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
643.0ms
(* (* (/ 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))))))

prune34.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.9b

localize74.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.9b
(pow (/ 1 (fabs x)) 4)
0.5b
(* (/ 1 (fabs x)) (/ 1 (fabs x)))

rewrite14.0ms

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
9.0ms
(* (/ 1 (fabs x)) (/ 1 (fabs x)))
3.0ms
(pow (/ 1 (fabs x)) 4)
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series63.0ms

Counts
4 → 8
Calls
4 calls:
Slowest
32.0ms
(pow (/ 1 (fabs x)) 4)
31.0ms
(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify747.0ms

Counts
48 → 84
Calls
48 calls:
Slowest
217.0ms
(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))))
195.0ms
(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
136.0ms
(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))))
26.0ms
(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))
20.0ms
(+ (- (log (fabs x))) (log (/ 1 (fabs x))))

prune2.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.1b

localize25.0ms

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)))

rewrite11.0ms

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

series41.0ms

Counts
4 → 8
Calls
4 calls:
Slowest
24.0ms
(* (/ 1 (fabs x)) (/ 1 (fabs x)))
17.0ms
(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.0s

Counts
70 → 102
Calls
70 calls:
Slowest
526.0ms
(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
171.0ms
(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
40.0ms
(+ (- (log (fabs x))) (log (/ 1 (fabs x))))
31.0ms
(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))
26.0ms
(+ (- (log (fabs x))) (log (/ 1 (fabs x))))

prune2.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.1b

localize31.0ms

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)))

rewrite21.0ms

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)))
9.0ms
(* (/ 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)))
23.0ms
(* (/ 1 (fabs x)) (/ 1 (fabs x)))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify1.0s

Counts
70 → 102
Calls
70 calls:
Slowest
225.0ms
(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
202.0ms
(fma (/ (exp (* (fabs x) (fabs x))) (sqrt PI)) (fma (* (* 3/4 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x)))) (pow (fabs x) -2) (fma (* 1/2 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x)))) (* (* (/ 1 (pow (fabs x) 3)) (pow (fabs x) (* -1 4))) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8))))
183.0ms
(* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))
130.0ms
(fma (/ (exp (* (fabs x) (fabs x))) (sqrt PI)) (fma (* (* 3/4 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x)))) (pow (fabs x) -2) (fma (* 1/2 (/ 1 (fabs x))) (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x)))) (* (* (/ 1 (pow (fabs x) 3)) (pow (fabs x) (* -1 4))) (* (exp (* (fabs x) (fabs x))) (* (/ 1 (sqrt PI)) 15/8))))
32.0ms
(+ (log (/ 1 (fabs x))) (log (/ 1 (fabs x))))

prune2.8s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.1b

localize38.0ms

Local error

Found 4 expressions with local error:

1.0b
(sqrt PI)
1.0b
(sqrt PI)
0.4b
(* (/ (/ 1 (fabs x)) (* (fabs x) (fabs x))) (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4)))
0.4b
(* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))

rewrite40.0ms

Algorithm
rewrite-expression-head
Rules
83×add-exp-log
46×prod-exp
27×add-cbrt-cube
16×div-exp
12×cbrt-unprod
12×associate-*l*
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
8×rec-exp
7×associate-*r*
6×pow-exp
6×pow1
6×pow-to-exp
4×add-log-exp
4×log1p-expm1-u
4×associate-*l/
4×times-frac
4×cbrt-undiv
4×expm1-log1p-u
3×div-inv
3×unpow-prod-down
2×*-commutative
2×pow1/2
1×associate-*r/
1×pow-prod-down
1×frac-times
Counts
4 → 94
Calls
4 calls:
Slowest
29.0ms
(* (/ (/ 1 (fabs x)) (* (fabs x) (fabs x))) (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4)))
9.0ms
(* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

series343.0ms

Counts
4 → 8
Calls
4 calls:
Slowest
202.0ms
(* (/ (/ 1 (fabs x)) (* (fabs x) (fabs x))) (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4)))
141.0ms
(* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))
0.0ms
(sqrt PI)
0.0ms
(sqrt PI)

simplify7.4s

Counts
60 → 102
Calls
60 calls:
Slowest
597.0ms
(+ (- (- (log (fabs x))) (log (* (fabs x) (fabs x)))) (log (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))))
515.0ms
(+ (- (- (log (fabs x))) (+ (log (fabs x)) (log (fabs x)))) (log (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))))
322.0ms
(+ (log (/ (/ 1 (fabs x)) (* (fabs x) (fabs x)))) (log (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))))
311.0ms
(* (/ (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x)))) (* (* (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4)) (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))) (* (/ 15/8 (sqrt PI)) (pow (fabs x) -4))))
276.0ms
(* (/ (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (* (* (* (fabs x) (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x)))) (* (* (* (/ 15/8 (sqrt PI)) (/ 15/8 (sqrt PI))) (/ 15/8 (sqrt PI))) (* (* (pow (fabs x) -4) (pow (fabs x) -4)) (pow (fabs x) -4))))

prune2.1s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0b

regimes270.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch3.0ms

end0.0ms

sample48.3s

Algorithm
intervals