Average Error: 13.9 → 0.3
Time: 19.9s
Precision: 64
Internal Precision: 128
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{(x \cdot x + x)_*}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.9

    \[\frac{1}{x + 1} - \frac{1}{x}\]

  2. Initial simplification13.9

    \[\leadsto \frac{1}{x + 1} - \frac{1}{x}\]
  3. Using strategy rm

  4. Applied frac-sub13.3

    \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]

  5. Simplified0.3

    \[\leadsto \frac{\color{blue}{-1}}{\left(x + 1\right) \cdot x}\]

  6. Simplified0.3

    \[\leadsto \frac{-1}{\color{blue}{(x \cdot x + x)_*}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

    \[\leadsto \frac{-1}{\color{blue}{1 \cdot (x \cdot x + x)_*}}\]

  9. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{-1}{1}}{(x \cdot x + x)_*}}\]

  10. Final simplification0.3

    \[\leadsto \frac{-1}{(x \cdot x + x)_*}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))

Details

Time bar (total: 15.2s)Debug log

start86.0ms

Algorithm
intervals

setup47.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 12.5b

localize22.0ms

Local error

Found 2 expressions with local error:

1.8b
(- (/ 1 (+ x 1)) (/ 1 x))
0.0b
(/ 1 (+ x 1))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
2 → 70
Calls

2 calls. Slowest were:

12.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
2.0ms
(/ 1 (+ x 1))

series29.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

18.0ms
(- (/ 1 (+ x 1)) (/ 1 x))
10.0ms
(/ 1 (+ x 1))

simplify6.4s

Counts
73 → 76
Calls

73 calls. Slowest were:

984.0ms
(fma (/ 1 (- (* x x) (* 1 1))) (- x 1) (- (* (sqrt (/ 1 x)) (sqrt (/ 1 x)))))
964.0ms
(fma (/ 1 (- (* x x) (* 1 1))) (- x 1) (- (* (/ 1 x) 1)))
964.0ms
(fma (/ 1 (- (* x x) (* 1 1))) (- x 1) (- (* (/ 1 x) 1)))

prune710.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0.3b

localize13.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ -1 (fma x x x))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
1 → 15
Calls

1 calls. Slowest were:

1.0ms
(/ -1 (fma x x x))

series20.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

19.0ms
(/ -1 (fma x x x))

simplify151.0ms

Counts
6 → 18
Calls

6 calls. Slowest were:

67.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
54.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))
26.0ms
(- 1 (+ x (/ 1 x)))

prune171.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.2b

localize11.0ms

Local error

Found 1 expressions with local error:

0.3b
(/ (/ -1 1) (fma x x x))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
1 → 34
Calls

1 calls. Slowest were:

9.0ms
(/ (/ -1 1) (fma x x x))

series18.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

18.0ms
(/ (/ -1 1) (fma x x x))

simplify1.7s

Counts
34 → 37
Calls

34 calls. Slowest were:

1.1s
(/ (* (* (/ -1 1) (/ -1 1)) (/ -1 1)) (* (* (fma x x x) (fma x x x)) (fma x x x)))
49.0ms
(- (log (/ -1 1)) (log (fma x x x)))
45.0ms
(- (/ 1 (pow x 3)) (+ (/ 1 (pow x 4)) (/ 1 (pow x 2))))

prune332.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.2b

localize17.0ms

Local error

Found 4 expressions with local error:

14.8b
(cbrt (/ -1 (fma x x x)))
14.8b
(cbrt (/ -1 (fma x x x)))
14.8b
(cbrt (/ -1 (fma x x x)))
0.5b
(* (cbrt (/ -1 (fma x x x))) (cbrt (/ -1 (fma x x x))))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 80
Calls

4 calls. Slowest were:

9.0ms
(* (cbrt (/ -1 (fma x x x))) (cbrt (/ -1 (fma x x x))))
1.0ms
(cbrt (/ -1 (fma x x x)))
1.0ms
(cbrt (/ -1 (fma x x x)))

series337.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

103.0ms
(* (cbrt (/ -1 (fma x x x))) (cbrt (/ -1 (fma x x x))))
82.0ms
(cbrt (/ -1 (fma x x x)))
78.0ms
(cbrt (/ -1 (fma x x x)))
74.0ms
(cbrt (/ -1 (fma x x x)))

simplify3.8s

Counts
53 → 92
Calls

53 calls. Slowest were:

375.0ms
(- (+ (* 5/9 (* (pow (pow x 4) 1/3) (pow (cbrt -1) 2))) (* (pow (/ 1 (pow x 2)) 1/3) (pow (cbrt -1) 2))) (* 2/3 (* (pow x 1/3) (pow (cbrt -1) 2))))
353.0ms
(- (+ (* 5/9 (* (pow (/ 1 (pow x 10)) 1/3) (pow (cbrt -1) 2))) (* (pow (/ 1 (pow x 4)) 1/3) (pow (cbrt -1) 2))) (* 2/3 (* (pow (/ 1 (pow x 7)) 1/3) (pow (cbrt -1) 2))))
307.0ms
(- (+ (* 2/9 (* (pow (pow x 5) 1/3) (cbrt -1))) (* (pow (/ 1 x) 1/3) (cbrt -1))) (* 1/3 (* (pow (pow x 2) 1/3) (cbrt -1))))

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.2b

regimes34.0ms

Accuracy

0% (0.1b remaining)

Error of 0.3b against oracle of 0.3b and baseline of 0.3b

bsearch1.0ms