Average Error: 32.2 → 0
Time: 4.3s
Precision: 64
Internal Precision: 128
\[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]
↓
\[1 - \frac{\left|x\right|}{x}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.2
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Derivation

  1. Initial program 32.2

    \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x}\]

  2. Simplified31.1

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

  3. Taylor expanded around 0 0

    \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}}\]

  4. Final simplification0

    \[\leadsto 1 - \frac{\left|x\right|}{x}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"

  :herbie-target
  (if (< x 0) 2 0)

  (- (/ x x) (* (/ 1 x) (sqrt (* x x)))))

Details

Time bar (total: 4.1s)Debug log

sample91.0ms

Algorithm
intervals
Results
256×(pre true 80)
139×(body real 80)
117×(body real 1280)

simplify63.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
63.0ms
(- (/ x x) (* (/ 1 x) (sqrt (* x x))))

prune10.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 16.9b

localize18.0ms

Local error

Found 1 expressions with local error:

28.2b
(fma (/ -1 x) (fabs x) 1)

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×fma-udef
1×add-cube-cbrt
1×add-exp-log
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 10
Calls
1 calls:
Slowest
0.0ms
(fma (/ -1 x) (fabs x) 1)

series32.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
32.0ms
(fma (/ -1 x) (fabs x) 1)

simplify13.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
4.0ms
(- 1 (/ (fabs x) x))
4.0ms
(- 1 (/ (fabs x) x))
4.0ms
(- 1 (/ (fabs x) x))

prune77.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize17.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Rules
Counts
0 → 0
Calls
0 calls:
Slowest

series0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

simplify0.0ms

Counts
0 → 0
Calls
0 calls:
Slowest

prune3.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

end0.0ms

sample3.8s

Algorithm
intervals
Results
8000×(pre true 80)
4012×(body real 1280)
3988×(body real 80)