Average Error: 58.1 → 0.6
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\sqrt{e^{-x}} + \sqrt{e^{x}}\right) \cdot \left(\left({x}^{3} \cdot \frac{1}{24} + \frac{1}{1920} \cdot {x}^{5}\right) + x\right)}{2}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

    \[\frac{e^{x} - e^{-x}}{2}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt58.1

    \[\leadsto \frac{e^{x} - \color{blue}{\sqrt{e^{-x}} \cdot \sqrt{e^{-x}}}}{2}\]

  4. Applied add-sqr-sqrt58.1

    \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - \sqrt{e^{-x}} \cdot \sqrt{e^{-x}}}{2}\]

  5. Applied difference-of-squares58.1

    \[\leadsto \frac{\color{blue}{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{e^{-x}}\right)}}{2}\]

  6. Taylor expanded around 0 0.6

    \[\leadsto \frac{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \color{blue}{\left(x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{1920} \cdot {x}^{5}\right)\right)}}{2}\]

  7. Final simplification0.6

    \[\leadsto \frac{\left(\sqrt{e^{-x}} + \sqrt{e^{x}}\right) \cdot \left(\left({x}^{3} \cdot \frac{1}{24} + \frac{1}{1920} \cdot {x}^{5}\right) + x\right)}{2}\]

Reproduce

herbie shell --seed 2019022 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))

Details

Time bar (total: 1.1m)Debug log

sample150.0ms

Algorithm
intervals
Results
69.0ms113×body1280valid
30.0ms80×body640valid
24.0ms249×body80valid
10.0ms38×body320valid
2.0ms15×body160valid

simplify37.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
37.0ms
(/ (- (exp x) (exp (- x))) 2)

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 57.8b

localize24.0ms

Local error

Found 1 expressions with local error:

6.4b
(- (exp x) (exp (- x)))

rewrite21.0ms

Algorithm
rewrite-expression-head
Rules
10×*-un-lft-identity
10×add-sqr-sqrt
9×prod-diff
8×add-cube-cbrt
3×add-log-exp
3×fma-neg
1×difference-of-squares
1×distribute-lft-out--
1×flip--
1×log1p-expm1-u
1×add-exp-log
1×diff-log
1×add-cbrt-cube
1×flip3--
1×sub-neg
1×pow1
1×expm1-log1p-u
Counts
1 → 28
Calls
1 calls:
Slowest
20.0ms
(- (exp x) (exp (- x)))

series41.0ms

Counts
1 → 3
Calls
1 calls:
Slowest
41.0ms
(- (exp x) (exp (- x)))

simplify669.0ms

Counts
27 → 31
Calls
27 calls:
Slowest
89.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
57.0ms
(fma 1 (exp x) (- (* (exp (- x)) 1)))
50.0ms
(fma 1 (exp x) (- (* (cbrt (exp (- x))) (* (cbrt (exp (- x))) (cbrt (exp (- x)))))))
38.0ms
(fma (- (cbrt (exp (- x)))) (* (cbrt (exp (- x))) (cbrt (exp (- x)))) (* (cbrt (exp (- x))) (* (cbrt (exp (- x))) (cbrt (exp (- x))))))
37.0ms
(fma (- (cbrt (exp (- x)))) (* (cbrt (exp (- x))) (cbrt (exp (- x)))) (* (cbrt (exp (- x))) (* (cbrt (exp (- x))) (cbrt (exp (- x))))))

prune260.0ms

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

Local error

Found 3 expressions with local error:

0.0b
(* (pow x 5) 1/60)
0.0b
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
0.0b
(fma 1/3 (* x x) 2)

rewrite3.0ms

Algorithm
rewrite-expression-head
Rules
4×add-cube-cbrt
4×*-un-lft-identity
4×add-sqr-sqrt
3×add-log-exp
3×log1p-expm1-u
3×add-exp-log
3×associate-*l*
3×add-cbrt-cube
3×pow1
3×expm1-log1p-u
2×fma-udef
1×*-commutative
Counts
3 → 33
Calls
3 calls:
Slowest
2.0ms
(* (pow x 5) 1/60)
0.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
0.0ms
(fma 1/3 (* x x) 2)

series108.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
43.0ms
(fma (fma 1/3 (* x x) 2) x (* (pow x 5) 1/60))
34.0ms
(* (pow x 5) 1/60)
31.0ms
(fma 1/3 (* x x) 2)

simplify298.0ms

Counts
11 → 42
Calls
11 calls:
Slowest
90.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
89.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
77.0ms
(+ (* 2 x) (+ (* 1/3 (pow x 3)) (* 1/60 (pow x 5))))
13.0ms
(+ (* 1/3 (pow x 2)) 2)
13.0ms
(+ (* 1/3 (pow x 2)) 2)

prune411.0ms

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0.0b

localize25.0ms

Local error

Found 4 expressions with local error:

6.4b
(- (sqrt (exp x)) (sqrt (exp (- x))))
0.0b
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
0.0b
(+ (sqrt (exp x)) (sqrt (exp (- x))))
0.0b
(sqrt (exp x))

rewrite72.0ms

Algorithm
rewrite-expression-head
Rules
133×sqrt-prod
108×prod-diff
100×add-sqr-sqrt
99×*-un-lft-identity
83×add-cube-cbrt
38×distribute-lft-in
38×distribute-rgt-in
9×associate-*r*
8×difference-of-squares
8×add-log-exp
6×add-exp-log
6×fma-neg
6×add-cbrt-cube
6×fma-def
6×pow1
5×associate-*l*
4×distribute-lft-out--
4×distribute-lft-out
4×flip--
4×log1p-expm1-u
4×flip-+
4×flip3--
4×flip3-+
4×frac-times
4×expm1-log1p-u
3×sub-neg
2×associate-*r/
2×associate-*l/
1×cbrt-unprod
1×*-commutative
1×prod-exp
1×pow-prod-down
1×diff-log
1×pow1/2
1×sum-log
1×rem-sqrt-square
1×+-commutative
Counts
4 → 208
Calls
4 calls:
Slowest
43.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
18.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
6.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))
1.0ms
(sqrt (exp x))

series332.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
188.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (- (sqrt (exp x)) (sqrt (exp (- x)))))
74.0ms
(- (sqrt (exp x)) (sqrt (exp (- x))))
56.0ms
(+ (sqrt (exp x)) (sqrt (exp (- x))))
13.0ms
(sqrt (exp x))

simplify52.3s

Counts
217 → 220
Calls
217 calls:
Slowest
1.5s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))
1.5s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))
1.5s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))
1.4s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))
1.3s
(* (fma (- (sqrt (cbrt (exp (- x))))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))) (* (sqrt (cbrt (exp (- x)))) (sqrt (* (cbrt (exp (- x))) (cbrt (exp (- x))))))) (+ (sqrt (exp x)) (sqrt (exp (- x)))))

prune2.4s

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0.0b

localize19.0ms

Local error

Found 4 expressions with local error:

0.1b
(* 1/24 (pow x 3))
0.0b
(* 1/1920 (pow x 5))
0.0b
(+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))
0.0b
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))))

rewrite26.0ms

Algorithm
rewrite-expression-head
Rules
12×*-un-lft-identity
11×associate-*r*
8×add-cube-cbrt
8×add-sqr-sqrt
7×flip-+
7×flip3-+
6×add-log-exp
6×add-exp-log
6×add-cbrt-cube
6×pow1
5×associate-*l*
4×log1p-expm1-u
4×frac-times
4×expm1-log1p-u
3×*-commutative
2×distribute-lft-out
2×sqrt-prod
2×associate-*r/
2×associate-*l/
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×unpow3
1×distribute-lft-in
1×fma-def
1×sum-log
1×+-commutative
1×cube-mult
1×distribute-rgt-in
Counts
4 → 73
Calls
4 calls:
Slowest
16.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))))
5.0ms
(* 1/24 (pow x 3))
4.0ms
(+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))
1.0ms
(* 1/1920 (pow x 5))

series233.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
116.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))))
68.0ms
(+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))
31.0ms
(* 1/1920 (pow x 5))
18.0ms
(* 1/24 (pow x 3))

simplify3.6s

Counts
43 → 85
Calls
43 calls:
Slowest
452.0ms
(* (- (* (sqrt (exp x)) (sqrt (exp x))) (* (sqrt (exp (- x))) (sqrt (exp (- x))))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))))
394.0ms
(* (* (* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (+ (sqrt (exp x)) (sqrt (exp (- x))))) (+ (sqrt (exp x)) (sqrt (exp (- x))))) (* (* (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5))))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5))))))
335.0ms
(* (+ (pow (sqrt (exp x)) 3) (pow (sqrt (exp (- x))) 3)) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))))
263.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (+ x (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5)))))
242.0ms
(* (+ (sqrt (exp x)) (sqrt (exp (- x)))) (+ (* 1/24 (pow x 3)) (* 1/1920 (pow x 5))))

prune1.4s

Pruning

4 alts after pruning (2 fresh and 2 done)

Merged error: 0.0b

regimes28.0ms

Accuracy

0% (0.5b remaining)

Error of 0.6b against oracle of 0.1b and baseline of 0.6b

bsearch2.0ms

end0.0ms

sample5.2s

Algorithm
intervals
Results
2.3s3384×body1280valid
1.4s2457×body640valid
775.0ms8087×body80valid
372.0ms1296×body320valid
132.0ms598×body160valid