Average Error: 61.8 → 0.3
Time: 5.0s
Precision: 64
Internal Precision: 128
\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\]
↓
\[{t}^{2} \cdot 3.9999999999999997 \cdot 10^{-32}\]

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. Simplified57.6

    \[\leadsto \color{blue}{(-2 \cdot \left(2 \cdot 10^{-16} \cdot t\right) + \left((\left((t \cdot \left( 2 \cdot 10^{-16} \right) + 1)_*\right) \cdot \left((t \cdot \left( 2 \cdot 10^{-16} \right) + 1)_*\right) + -1)_*\right))_*}\]

  3. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{3.9999999999999997 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Final simplification0.3

    \[\leadsto {t}^{2} \cdot 3.9999999999999997 \cdot 10^{-32}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(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: 4.3s)Debug log

sample48.0ms

Algorithm
intervals
Results
22.0ms194×body160valid
11.0ms62×body320valid
5.0ms256×pre80true

simplify307.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
306.0ms
(+ (* (+ 1 (* t 2e-16)) (+ 1 (* t 2e-16))) (- -1 (* 2 (* t 2e-16))))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.6b

localize28.0ms

Local error

Found 2 expressions with local error:

52.9b
(fma -2 (* 2e-16 t) (fma (fma t 2e-16 1) (fma t 2e-16 1) -1))
49.6b
(fma (fma t 2e-16 1) (fma t 2e-16 1) -1)

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
2×log1p-expm1-u
2×fma-udef
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
2×expm1-log1p-u
2×add-sqr-sqrt
Counts
2 → 20
Calls
2 calls:
Slowest
0.0ms
(fma -2 (* 2e-16 t) (fma (fma t 2e-16 1) (fma t 2e-16 1) -1))
0.0ms
(fma (fma t 2e-16 1) (fma t 2e-16 1) -1)

series154.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
88.0ms
(fma -2 (* 2e-16 t) (fma (fma t 2e-16 1) (fma t 2e-16 1) -1))
66.0ms
(fma (fma t 2e-16 1) (fma t 2e-16 1) -1)

simplify80.0ms

Counts
6 → 26
Calls
6 calls:
Slowest
24.0ms
(+ (* 4e-16 t) (* 3.9999999999999997e-32 (pow t 2)))
24.0ms
(+ (* 4e-16 t) (* 3.9999999999999997e-32 (pow t 2)))
15.0ms
(+ (* 4e-16 t) (* 3.9999999999999997e-32 (pow t 2)))
7.0ms
(* 3.9999999999999997e-32 (pow t 2))
5.0ms
(* 3.9999999999999997e-32 (pow t 2))

prune199.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize12.0ms

Local error

Found 1 expressions with local error:

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

rewrite2.0ms

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×log1p-expm1-u
1×*-commutative
1×add-exp-log
1×add-cbrt-cube
1×pow1
1×expm1-log1p-u
Counts
1 → 14
Calls
1 calls:
Slowest
2.0ms
(* 3.9999999999999997e-32 (pow t 2))

series23.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
23.0ms
(* 3.9999999999999997e-32 (pow t 2))

simplify29.0ms

Counts
6 → 17
Calls
6 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
(cbrt (pow t 2))
3.0ms
(sqrt (pow t 2))

prune106.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0.1b

localize5.0ms

Local error

Found 1 expressions with local error:

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

rewrite2.0ms

Algorithm
rewrite-expression-head
Rules
1×add-log-exp
1×log1p-expm1-u
1×add-cube-cbrt
1×*-commutative
1×add-exp-log
1×associate-*l*
1×add-cbrt-cube
1×*-un-lft-identity
1×pow1
1×expm1-log1p-u
1×add-sqr-sqrt
Counts
1 → 11
Calls
1 calls:
Slowest
2.0ms
(* (* 3.9999999999999997e-32 t) t)

series12.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
12.0ms
(* (* 3.9999999999999997e-32 t) t)

simplify13.0ms

Counts
3 → 14
Calls
3 calls:
Slowest
6.0ms
(* 3.9999999999999997e-32 (pow t 2))
4.0ms
(* 3.9999999999999997e-32 (pow t 2))
3.0ms
(* 3.9999999999999997e-32 (pow t 2))

prune114.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.1b

localize10.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (sqrt (* (* 3.9999999999999997e-32 t) t)) (sqrt (* (* 3.9999999999999997e-32 t) t)))
0.3b
(* (* 3.9999999999999997e-32 t) t)
0.3b
(* (* 3.9999999999999997e-32 t) t)
0.1b
(sqrt (* (* 3.9999999999999997e-32 t) t))

rewrite22.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
6×add-cube-cbrt
6×add-exp-log
6×associate-*l*
6×add-cbrt-cube
6×*-un-lft-identity
6×add-sqr-sqrt
5×pow1/2
4×add-log-exp
4×log1p-expm1-u
4×associate-*r*
4×expm1-log1p-u
3×sqrt-prod
3×*-commutative
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×rem-square-sqrt
1×pow2
Counts
4 → 61
Calls
4 calls:
Slowest
15.0ms
(* (sqrt (* (* 3.9999999999999997e-32 t) t)) (sqrt (* (* 3.9999999999999997e-32 t) t)))
3.0ms
(* (* 3.9999999999999997e-32 t) t)
2.0ms
(* (* 3.9999999999999997e-32 t) t)
1.0ms
(sqrt (* (* 3.9999999999999997e-32 t) t))

series131.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
60.0ms
(* (sqrt (* (* 3.9999999999999997e-32 t) t)) (sqrt (* (* 3.9999999999999997e-32 t) t)))
33.0ms
(sqrt (* (* 3.9999999999999997e-32 t) t))
19.0ms
(* (* 3.9999999999999997e-32 t) t)
18.0ms
(* (* 3.9999999999999997e-32 t) t)

simplify592.0ms

Counts
31 → 73
Calls
31 calls:
Slowest
325.0ms
(* (* (* (sqrt (* (* 3.9999999999999997e-32 t) t)) (sqrt (* (* 3.9999999999999997e-32 t) t))) (sqrt (* (* 3.9999999999999997e-32 t) t))) (* (* (sqrt (* (* 3.9999999999999997e-32 t) t)) (sqrt (* (* 3.9999999999999997e-32 t) t))) (sqrt (* (* 3.9999999999999997e-32 t) t))))
64.0ms
(* (* (* 3.9999999999999997e-32 t) t) (* (* 3.9999999999999997e-32 t) t))
43.0ms
(* (* (* 3.9999999999999997e-32 t) t) (* (* 3.9999999999999997e-32 t) t))
12.0ms
(* -1 (* t (sqrt 3.9999999999999997e-32)))
11.0ms
(cbrt (sqrt (* (* 3.9999999999999997e-32 t) t)))

prune553.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes41.0ms

Accuracy

0% (0.3b remaining)

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

bsearch1.0ms

end0.0ms

sample1.8s

Algorithm
intervals
Results
939.0ms6003×body160valid
538.0ms1997×body320valid
246.0ms8000×pre80true