Average Error: 29.8 → 0.6
Time: 11.1s
Precision: 64
Internal Precision: 128
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*\]

Error

Bits error versus x

Target

Original29.8
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

    \[\left(e^{x} - 2\right) + e^{-x}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*}\]

  4. Final simplification0.6

    \[\leadsto (\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))

Details

Time bar (total: 10.6s)Debug log

sample132.0ms

Algorithm
intervals
Results
94.0ms188×body1280valid
12.0ms246×body80valid
9.0ms30×body640valid
5.0ms24×body320valid
1.0ms7×body160valid

simplify46.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
46.0ms
(+ (- (exp x) 2) (exp (- x)))

prune5.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 27.1b

localize11.0ms

Local error

Found 2 expressions with local error:

4.4b
(+ (- (exp x) 2) (exp (- x)))
0.0b
(- (exp x) 2)

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
6×*-un-lft-identity
4×add-log-exp
4×add-cube-cbrt
4×frac-add
4×add-sqr-sqrt
3×flip--
3×fma-neg
3×flip3--
3×fma-def
2×log1p-expm1-u
2×add-exp-log
2×exp-diff
2×add-cbrt-cube
2×sub-neg
2×associate-+l+
2×pow1
2×expm1-log1p-u
2×exp-neg
2×neg-sub0
1×distribute-lft-out
1×flip-+
1×associate-+l-
1×flip3-+
1×sum-log
1×+-commutative
Counts
2 → 40
Calls
2 calls:
Slowest
14.0ms
(+ (- (exp x) 2) (exp (- x)))
5.0ms
(- (exp x) 2)

series108.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
95.0ms
(+ (- (exp x) 2) (exp (- x)))
13.0ms
(- (exp x) 2)

simplify1.4s

Counts
23 → 46
Calls
23 calls:
Slowest
409.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp 0)))
224.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) (exp 0)))
185.0ms
(+ (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) 1))
181.0ms
(+ (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) 1))
123.0ms
(* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) (exp x))

prune395.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 2 expressions with local error:

0.0b
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))
0.0b
(fma 1/360 (pow x 6) (* x x))

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 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))
0.0ms
(fma 1/360 (pow x 6) (* x x))

series99.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
69.0ms
(fma 1/360 (pow x 6) (* x x))
30.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

simplify63.0ms

Counts
6 → 26
Calls
6 calls:
Slowest
20.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
15.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
13.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
5.0ms
(+ (pow x 2) (* 1/360 (pow x 6)))
5.0ms
(+ (pow x 2) (* 1/360 (pow x 6)))

prune235.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize10.0ms

Local error

Found 4 expressions with local error:

30.5b
(sqrt (fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x))))
30.5b
(sqrt (fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x))))
0.0b
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))
0.0b
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
8×add-sqr-sqrt
6×sqrt-prod
6×add-cube-cbrt
6×*-un-lft-identity
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×pow1
4×expm1-log1p-u
2×fma-udef
2×pow1/2
2×rem-sqrt-square
Counts
4 → 48
Calls
4 calls:
Slowest
1.0ms
(sqrt (fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x))))
1.0ms
(sqrt (fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x))))
0.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))
0.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

series135.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
45.0ms
(sqrt (fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x))))
41.0ms
(sqrt (fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x))))
27.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))
22.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

simplify1.3s

Counts
20 → 60
Calls
20 calls:
Slowest
256.0ms
(- (+ (* 1/2 (/ 1 (* (sqrt 1/360) x))) (+ (* (sqrt 1/360) (pow x 3)) (* 1/24 (/ x (sqrt 1/360))))) (* 1/1152 (/ 1 (* (pow (sqrt 1/360) 3) x))))
251.0ms
(- (+ (* 1/2 (/ 1 (* (sqrt 1/360) x))) (+ (* (sqrt 1/360) (pow x 3)) (* 1/24 (/ x (sqrt 1/360))))) (* 1/1152 (/ 1 (* (pow (sqrt 1/360) 3) x))))
209.0ms
(- (* 1/1152 (/ 1 (* (pow (sqrt 1/360) 3) x))) (+ (* 1/2 (/ 1 (* (sqrt 1/360) x))) (+ (* (sqrt 1/360) (pow x 3)) (* 1/24 (/ x (sqrt 1/360))))))
179.0ms
(- (* 1/1152 (/ 1 (* (pow (sqrt 1/360) 3) x))) (+ (* 1/2 (/ 1 (* (sqrt 1/360) x))) (+ (* (sqrt 1/360) (pow x 3)) (* 1/24 (/ x (sqrt 1/360))))))
51.0ms
(+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5))))

prune725.0ms

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 2 expressions with local error:

2.0b
(- (+ (exp x) (exp (- x))) 2)
0.0b
(+ (exp x) (exp (- x)))

rewrite54.0ms

Algorithm
rewrite-expression-head
Rules
8×*-un-lft-identity
4×add-log-exp
4×add-cube-cbrt
4×fma-neg
4×add-sqr-sqrt
3×fma-def
2×distribute-lft-out
2×log1p-expm1-u
2×add-exp-log
2×add-cbrt-cube
2×pow1
2×expm1-log1p-u
1×flip--
1×flip-+
1×associate--l+
1×flip3--
1×flip3-+
1×sub-neg
1×sum-log
1×+-commutative
Counts
2 → 35
Calls
2 calls:
Slowest
49.0ms
(- (+ (exp x) (exp (- x))) 2)
5.0ms
(+ (exp x) (exp (- x)))

series103.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
71.0ms
(- (+ (exp x) (exp (- x))) 2)
32.0ms
(+ (exp x) (exp (- x)))

simplify114.0ms

Counts
13 → 41
Calls
13 calls:
Slowest
24.0ms
(- (+ (exp (* -1 x)) (exp x)) 2)
18.0ms
(- (+ (exp x) (exp (- x))) 2)
13.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
13.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) 2))
9.0ms
(+ (exp (* -1 x)) (exp x))

prune243.0ms

Pruning

4 alts after pruning (1 fresh and 3 done)

Merged error: 0.0b

regimes26.0ms

Accuracy

0% (0.6b remaining)

Error of 0.6b against oracle of 0.0b and baseline of 0.6b

bsearch1.0ms

end0.0ms

sample5.4s

Algorithm
intervals
Results
3.6s5636×body1280valid
896.0ms1279×body640valid
482.0ms7995×body80valid
154.0ms624×body320valid
37.0ms288×body160valid