Average Error: 61.8 → 0.3
Time: 5.2s
Precision: 64
Internal Precision: 128
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[\left(\left(2 \cdot 10^{-16} \cdot t\right) \cdot t\right) \cdot 2 \cdot 10^{-16}\]

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.3
\[(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right))_*\]

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. Simplified0.4

    \[\leadsto \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right) \cdot \left(2 \cdot 10^{-16} \cdot t\right)}\]
  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019022 
(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: 4.6s)Debug log

sample60.0ms

Algorithm
intervals
Results
28.0ms194×body160valid
14.0ms62×body320valid
6.0ms256×pre80true

simplify1.2s

Counts
1 → 1
Calls
1 calls:
Slowest
1.2s
(+ (* (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16))) (- -1 (* 2 (* t 2e-16))))

prune6.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize5.0ms

Local error

Found 1 expressions with local error:

0.4b
(* (* 2e-16 t) (* 2e-16 t))

rewrite4.0ms

Algorithm
rewrite-expression-head
Rules
6×pow1
3×add-exp-log
3×add-cbrt-cube
1×add-log-exp
1×cbrt-unprod
1×add-cube-cbrt
1×pow-prod-up
1×*-commutative
1×prod-exp
1×pow-prod-down
1×associate-*l*
1×associate-*r*
1×pow-plus
1×*-un-lft-identity
1×add-sqr-sqrt
1×pow2
Counts
1 → 16
Calls
1 calls:
Slowest
3.0ms
(* (* 2e-16 t) (* 2e-16 t))

series11.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
11.0ms
(* (* 2e-16 t) (* 2e-16 t))

simplify578.0ms

Counts
9 → 19
Calls
9 calls:
Slowest
538.0ms
(* (* (* (* 2e-16 t) (* 2e-16 t)) (* 2e-16 t)) (* (* (* 2e-16 t) (* 2e-16 t)) (* 2e-16 t)))
11.0ms
(+ (log (* 2e-16 t)) (log (* 2e-16 t)))
8.0ms
(* (* 2e-16 t) (* 2e-16 t))
4.0ms
(+ 1 1)
4.0ms
(* 3.9999999999999997e-32 (pow t 2))

prune120.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 2 expressions with local error:

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

rewrite8.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
2×add-cube-cbrt
2×*-commutative
2×add-exp-log
2×associate-*l*
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
2×add-sqr-sqrt
Counts
2 → 18
Calls
2 calls:
Slowest
6.0ms
(* (* (* 2e-16 t) 2e-16) t)
2.0ms
(* (* 2e-16 t) 2e-16)

series25.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
13.0ms
(* (* 2e-16 t) 2e-16)
12.0ms
(* (* (* 2e-16 t) 2e-16) t)

simplify33.0ms

Counts
7 → 24
Calls
7 calls:
Slowest
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
3.0ms
(* 3.9999999999999997e-32 t)
2.0ms
(* 2e-16 t)

prune118.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 2 expressions with local error:

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

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
2×add-cube-cbrt
2×*-commutative
2×add-exp-log
2×associate-*r*
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
2×add-sqr-sqrt
Counts
2 → 18
Calls
2 calls:
Slowest
4.0ms
(* 2e-16 (* t (* 2e-16 t)))
2.0ms
(* t (* 2e-16 t))

series41.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
22.0ms
(* 2e-16 (* t (* 2e-16 t)))
19.0ms
(* t (* 2e-16 t))

simplify44.0ms

Counts
7 → 24
Calls
7 calls:
Slowest
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
6.0ms
(* 2e-16 (pow t 2))
6.0ms
(* 2e-16 (pow t 2))

prune131.0ms

Pruning

8 alts after pruning (6 fresh and 2 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 1 expressions with local error:

0.3b
(* 3.9999999999999997e-32 (pow t 2))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
4×associate-*r*
2×add-cube-cbrt
2×*-un-lft-identity
2×add-sqr-sqrt
1×add-log-exp
1×unpow2
1×*-commutative
1×add-exp-log
1×add-cbrt-cube
1×pow1
Counts
1 → 12
Calls
1 calls:
Slowest
2.0ms
(* 3.9999999999999997e-32 (pow t 2))

series14.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
14.0ms
(* 3.9999999999999997e-32 (pow t 2))

simplify15.0ms

Counts
6 → 15
Calls
6 calls:
Slowest
4.0ms
(* 3.9999999999999997e-32 (pow t 2))
4.0ms
(* 3.9999999999999997e-32 (pow t 2))
3.0ms
(* 3.9999999999999997e-32 (pow t 2))
1.0ms
(cbrt (pow t 2))
1.0ms
(sqrt (pow t 2))

prune90.0ms

Pruning

9 alts after pruning (6 fresh and 3 done)

Merged error: 0.0b

regimes61.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.0b and baseline of 0.3b

bsearch2.0ms

end0.0ms

sample2.0s

Algorithm
intervals
Results
1.1s6003×body160valid
553.0ms1997×body320valid
190.0ms8000×pre80true