Average Error: 32.6 → 0
Time: 10.7s
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

Original32.6
Target0
Herbie0
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Derivation

  1. Initial program 32.6

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

  2. Initial simplification31.3

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

  3. Taylor expanded around 0 0

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

  4. Final simplification0

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

Runtime

Time bar (total: 10.7s)Debug logProfile

start293.0ms

Algorithm
intervals

setup331.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 19.1b

localize9.0ms

Local error

Found 1 expressions with local error:

31.0b
(fma (/ -1 x) (fabs x) 1)

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
1 → 10
Calls

1 calls. Slowest were:

0.0ms
(fma (/ -1 x) (fabs x) 1)

series19.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

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

simplify14.0ms

Counts
3 → 13
Calls

3 calls. Slowest were:

6.0ms
(- 1 (/ (fabs x) x))
4.0ms
(- 1 (/ (fabs x) x))
3.0ms
(- 1 (/ (fabs x) x))

prune119.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize10.0ms

Local error

Found 0 expressions with local error:

rewrite0.0ms

Algorithm
rewrite-expression-head
Counts
0 → 0
Calls

0 calls. Slowest were:

series0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

simplify0.0ms

Counts
0 → 0
Calls

0 calls. Slowest were:

prune7.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b

herbie shell --seed 2018357 +o rules:numerics
(FPCore (x)
  :name "sqrt sqr"

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

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