Average Error: 29.8 → 0.7
Time: 17.8s
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
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Initial simplification29.8

    \[\leadsto \left(e^{x} - 2\right) - \frac{-1}{e^{x}}\]

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2018362 +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: 11.5s)Debug log

start175.0ms

Algorithm
intervals

setup41.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.8b

localize11.0ms

Local error

Found 3 expressions with local error:

3.5b
(- (- (exp x) 2) (/ -1 (exp x)))
0.0b
(- (exp x) 2)
0.0b
(/ -1 (exp x))

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
3 → 67
Calls

3 calls. Slowest were:

42.0ms
(- (- (exp x) 2) (/ -1 (exp x)))
6.0ms
(- (exp x) 2)
1.0ms
(/ -1 (exp x))

series134.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

108.0ms
(- (- (exp x) 2) (/ -1 (exp x)))
13.0ms
(/ -1 (exp x))
13.0ms
(- (exp x) 2)

simplify2.6s

Counts
51 → 76
Calls

51 calls. Slowest were:

368.0ms
(fma (* (cbrt (- (exp x) 2)) (cbrt (- (exp x) 2))) (cbrt (- (exp x) 2)) (- (* (cbrt (/ -1 (exp x))) (* (cbrt (/ -1 (exp x))) (cbrt (/ -1 (exp x)))))))
343.0ms
(- (* (- (pow (exp x) 3) (pow 2 3)) (exp x)) (* (+ (* (exp x) (exp x)) (+ (* 2 2) (* (exp x) 2))) -1))
165.0ms
(- (* (- (* (exp x) (exp x)) (* 2 2)) (exp x)) (* (+ (exp x) 2) -1))

prune628.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

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
Counts
2 → 20
Calls

2 calls. Slowest were:

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))

series127.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

86.0ms
(fma 1/360 (pow x 6) (* x x))
40.0ms
(fma 1/12 (pow x 4) (fma 1/360 (pow x 6) (* x x)))

simplify67.0ms

Counts
6 → 26
Calls

6 calls. Slowest were:

20.0ms
(+ (pow x 2) (+ (* 1/12 (pow x 4)) (* 1/360 (pow x 6))))
16.0ms
(+ (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))))

prune383.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

localize42.0ms

Local error

Found 4 expressions with local error:

5.7b
(fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))
0.0b
(* (exp x) (exp x))
0.0b
(* (exp x) (exp x))
0.0b
(fma 2 (exp x) (fma (exp x) (exp x) 4))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

4.0ms
(* (exp x) (exp x))
4.0ms
(* (exp x) (exp x))
0.0ms
(fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))

series200.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

127.0ms
(fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))
41.0ms
(fma 2 (exp x) (fma (exp x) (exp x) 4))
16.0ms
(* (exp x) (exp x))
16.0ms
(* (exp x) (exp x))

simplify896.0ms

Counts
34 → 76
Calls

34 calls. Slowest were:

156.0ms
(- (+ (pow (exp x) 2) (+ (pow (exp x) 4) 4)) (* 6 (exp x)))
139.0ms
(+ (* 2 x) (+ (* 2 (pow x 2)) 1))
130.0ms
(- (+ (pow (exp x) 2) (+ (pow (exp x) 4) 4)) (* 6 (exp x)))

prune1.3s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

5.7b
(fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))
0.3b
(exp (log (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))))
0.0b
(* (exp x) (exp x))
0.0b
(* (exp x) (exp x))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

7.0ms
(* (exp x) (exp x))
7.0ms
(* (exp x) (exp x))
4.0ms
(exp (log (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))))

series279.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

140.0ms
(fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))
120.0ms
(exp (log (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))))
9.0ms
(* (exp x) (exp x))
9.0ms
(* (exp x) (exp x))

simplify2.8s

Counts
42 → 83
Calls

42 calls. Slowest were:

425.0ms
(exp (log (sqrt (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4))))))
416.0ms
(exp (* (cbrt (log (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4))))) (cbrt (log (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))))))
390.0ms
(exp (log (* (cbrt (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))) (cbrt (fma (fma (exp x) (* (exp x) (exp x)) -8) (exp x) (fma 2 (exp x) (fma (exp x) (exp x) 4)))))))

prune1.8s

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0.0b

regimes23.0ms

Accuracy

0% (0.6b remaining)

Error of 0.7b against oracle of 0.1b and baseline of 0.7b

bsearch1.0ms