Average Error: 61.8 → 0.4
Time: 6.1s
Precision: 64
Internal Precision: 128
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left({t}^{2} \cdot 2 \cdot 10^{-16}\right) \cdot 2 \cdot 10^{-16}\]

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.5
Herbie0.4
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]

  2. Initial simplification0.4

    \[\leadsto \left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)\]
  3. Using strategy rm

  4. Applied associate-*l*0.3

    \[\leadsto \color{blue}{2 \cdot 10^{-16} \cdot \left(t \cdot \left(2 \cdot 10^{-16} \cdot t\right)\right)}\]

  5. Taylor expanded around 0 0.4

    \[\leadsto 2 \cdot 10^{-16} \cdot \color{blue}{\left(2 \cdot 10^{-16} \cdot {t}^{2}\right)}\]

  6. Final simplification0.4

    \[\leadsto \left({t}^{2} \cdot 2 \cdot 10^{-16}\right) \cdot 2 \cdot 10^{-16}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16)) (- -1 (* 2 (* t 2e-16))))

  (+ (* (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16))) (- -1 (* 2 (* t 2e-16)))))

Details

Time bar (total: 2.6s)Debug log

start90.0ms

Algorithm
intervals

setup1.0s

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize12.0ms

Local error

Found 1 expressions with local error:

0.5b
(* (* 2e-16 t) (* 2e-16 t))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 16
Calls

1 calls. Slowest were:

7.0ms
(* (* 2e-16 t) (* 2e-16 t))

series20.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

20.0ms
(* (* 2e-16 t) (* 2e-16 t))

simplify682.0ms

Counts
9 → 19
Calls

9 calls. Slowest were:

610.0ms
(* (* (* (* 2e-16 t) (* 2e-16 t)) (* 2e-16 t)) (* (* (* 2e-16 t) (* 2e-16 t)) (* 2e-16 t)))
24.0ms
(+ (log (* 2e-16 t)) (log (* 2e-16 t)))
17.0ms
(* (* 2e-16 t) (* 2e-16 t))

prune132.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.1b

localize5.0ms

Local error

Found 2 expressions with local error:

0.3b
(* (* (* 2e-16 t) 2e-16) t)
0.3b
(* (* 2e-16 t) 2e-16)

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 18
Calls

2 calls. Slowest were:

5.0ms
(* (* (* 2e-16 t) 2e-16) t)
2.0ms
(* (* 2e-16 t) 2e-16)

series21.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

13.0ms
(* (* (* 2e-16 t) 2e-16) t)
7.0ms
(* (* 2e-16 t) 2e-16)

simplify17.0ms

Counts
7 → 24
Calls

7 calls. Slowest were:

4.0ms
(* 3.9999999999999997e-32 (pow t 2))
4.0ms
(* 3.9999999999999997e-32 (pow t 2))
4.0ms
(* 3.9999999999999997e-32 (pow t 2))

prune224.0ms

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 1 expressions with local error:

0.3b
(* (* 3.9999999999999997e-32 t) t)

rewrite2.0ms

Algorithm
rewrite-expression-head
Counts
1 → 9
Calls

1 calls. Slowest were:

2.0ms
(* (* 3.9999999999999997e-32 t) t)

series19.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

19.0ms
(* (* 3.9999999999999997e-32 t) t)

simplify15.0ms

Counts
3 → 12
Calls

3 calls. Slowest were:

7.0ms
(* 3.9999999999999997e-32 (pow t 2))
4.0ms
(* 3.9999999999999997e-32 (pow t 2))
4.0ms
(* 3.9999999999999997e-32 (pow t 2))

prune84.0ms

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 2 expressions with local error:

0.3b
(* 2e-16 (* t (* 2e-16 t)))
0.2b
(* t (* 2e-16 t))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
2 → 18
Calls

2 calls. Slowest were:

2.0ms
(* 2e-16 (* t (* 2e-16 t)))
1.0ms
(* t (* 2e-16 t))

series27.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

15.0ms
(* t (* 2e-16 t))
12.0ms
(* 2e-16 (* t (* 2e-16 t)))

simplify41.0ms

Counts
7 → 24
Calls

7 calls. Slowest were:

7.0ms
(* 3.9999999999999997e-32 (pow t 2))
6.0ms
(* 3.9999999999999997e-32 (pow t 2))
6.0ms
(* 3.9999999999999997e-32 (pow t 2))

prune138.0ms

Pruning

8 alts after pruning (5 fresh and 3 done)

Merged error: 0.0b

regimes46.0ms

Accuracy

0% (0.3b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch1.0ms