Average Error: 0.1 → 0.1
Time: 5.0s
Precision: 64
Internal Precision: 128
\[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\]
↓
\[(\left(-0.12900613773279798\right) \cdot \left({x}^{3}\right) + \left(x \cdot 0.954929658551372\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{(\left(-0.12900613773279798\right) \cdot \left({x}^{3}\right) + \left(0.954929658551372 \cdot x\right))_*}\]

  3. Final simplification0.1

    \[\leadsto (\left(-0.12900613773279798\right) \cdot \left({x}^{3}\right) + \left(x \cdot 0.954929658551372\right))_*\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "Rosa's Benchmark"
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

Details

Time bar (total: 4.7s)Debug log

sample167.0ms

Algorithm
intervals
Results
91.0ms367×body80valid

simplify106.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
97.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))

prune14.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 1 expressions with local error:

0.1b
(fma (- 0.12900613773279798) (pow x 3) (* 0.954929658551372 x))

rewrite3.0ms

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
1.0ms
(fma (- 0.12900613773279798) (pow x 3) (* 0.954929658551372 x))

series98.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
97.0ms
(fma (- 0.12900613773279798) (pow x 3) (* 0.954929658551372 x))

simplify157.0ms

Counts
3 → 13
Calls
3 calls:
Slowest
64.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))
51.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))
41.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))

prune148.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize20.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (* x x) x)
0.1b
(* 0.12900613773279798 (* (* x x) x))
0.0b
(- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
3×log1p-expm1-u
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
3×expm1-log1p-u
3×add-sqr-sqrt
2×*-commutative
1×flip--
1×pow3
1×prod-diff
1×associate-*l*
1×associate-*r*
1×diff-log
1×fma-neg
1×flip3--
1×pow-plus
1×sub-neg
1×pow2
Counts
3 → 40
Calls
3 calls:
Slowest
8.0ms
(* (* x x) x)
6.0ms
(* 0.12900613773279798 (* (* x x) x))
5.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))

series102.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
50.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x)))
28.0ms
(* 0.12900613773279798 (* (* x x) x))
24.0ms
(* (* x x) x)

simplify419.0ms

Counts
12 → 49
Calls
12 calls:
Slowest
94.0ms
(/ (exp (* 0.954929658551372 x)) (exp (* 0.12900613773279798 (* (* x x) x))))
67.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))
65.0ms
(fma 0.954929658551372 x (- (* (* (* x x) x) 0.12900613773279798)))
57.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))
51.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))

prune391.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0.0b

localize16.0ms

Local error

Found 3 expressions with local error:

0.1b
(* (* 0.12900613773279798 (* x x)) x)
0.1b
(* 0.12900613773279798 (* x x))
0.0b
(- (* 0.954929658551372 x) (* (* 0.12900613773279798 (* x x)) x))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
3×log1p-expm1-u
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
3×expm1-log1p-u
3×add-sqr-sqrt
2×*-commutative
1×flip--
1×prod-diff
1×associate-*l*
1×associate-*r*
1×diff-log
1×fma-neg
1×distribute-rgt-out--
1×flip3--
1×sub-neg
Counts
3 → 39
Calls
3 calls:
Slowest
7.0ms
(* (* 0.12900613773279798 (* x x)) x)
5.0ms
(- (* 0.954929658551372 x) (* (* 0.12900613773279798 (* x x)) x))
2.0ms
(* 0.12900613773279798 (* x x))

series94.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
49.0ms
(- (* 0.954929658551372 x) (* (* 0.12900613773279798 (* x x)) x))
31.0ms
(* (* 0.12900613773279798 (* x x)) x)
14.0ms
(* 0.12900613773279798 (* x x))

simplify481.0ms

Counts
13 → 48
Calls
13 calls:
Slowest
86.0ms
(fma 0.954929658551372 x (- (* x (* 0.12900613773279798 (* x x)))))
79.0ms
(/ (exp (* 0.954929658551372 x)) (exp (* (* 0.12900613773279798 (* x x)) x)))
74.0ms
(fma (- x) (* 0.12900613773279798 (* x x)) (* x (* 0.12900613773279798 (* x x))))
67.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))
65.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))

prune330.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.1b
(* (* 0.12900613773279798 x) x)
0.1b
(* (* (* 0.12900613773279798 x) x) x)
0.0b
(- (* 0.954929658551372 x) (* (* (* 0.12900613773279798 x) x) x))

rewrite11.0ms

Algorithm
rewrite-expression-head
Rules
5×add-log-exp
3×log1p-expm1-u
3×add-cube-cbrt
3×add-exp-log
3×add-cbrt-cube
3×*-un-lft-identity
3×pow1
3×expm1-log1p-u
3×add-sqr-sqrt
2×*-commutative
2×associate-*l*
1×flip--
1×prod-diff
1×diff-log
1×fma-neg
1×distribute-rgt-out--
1×flip3--
1×sub-neg
Counts
3 → 39
Calls
3 calls:
Slowest
5.0ms
(* (* (* 0.12900613773279798 x) x) x)
3.0ms
(* (* 0.12900613773279798 x) x)
2.0ms
(- (* 0.954929658551372 x) (* (* (* 0.12900613773279798 x) x) x))

series98.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
62.0ms
(- (* 0.954929658551372 x) (* (* (* 0.12900613773279798 x) x) x))
24.0ms
(* (* (* 0.12900613773279798 x) x) x)
12.0ms
(* (* 0.12900613773279798 x) x)

simplify343.0ms

Counts
14 → 48
Calls
14 calls:
Slowest
74.0ms
(/ (exp (* 0.954929658551372 x)) (exp (* (* (* 0.12900613773279798 x) x) x)))
57.0ms
(fma 0.954929658551372 x (- (* x (* (* 0.12900613773279798 x) x))))
49.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))
45.0ms
(fma (- x) (* (* 0.12900613773279798 x) x) (* x (* (* 0.12900613773279798 x) x)))
42.0ms
(- (* 0.954929658551372 x) (* 0.12900613773279798 (pow x 3)))

prune357.0ms

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

regimes38.0ms

Accuracy

0% (0.1b remaining)

Error of 0.1b against oracle of 0.0b and baseline of 0.1b

bsearch2.0ms

end0.0ms

sample1.2s

Algorithm
intervals
Results
1.1s11908×body80valid