Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
Internal Precision: 128
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied log1p-expm1-u0.0

    \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)}\]

  4. Final simplification0.0

    \[\leadsto \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))

Details

Time bar (total: 2.2s)Debug log

start45.0ms

Algorithm
intervals

setup27.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize39.0ms

Local error

Found 1 expressions with local error:

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

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
1 → 18
Calls

1 calls. Slowest were:

6.0ms
(/ 2 (+ (exp x) (exp (- x))))

series58.0ms

Counts
1 → 3
Calls

1 calls. Slowest were:

58.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify356.0ms

Counts
9 → 21
Calls

9 calls. Slowest were:

246.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
54.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))
16.0ms
(/ 2 (+ (exp (* -1 x)) (exp x)))

prune212.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 2 expressions with local error:

0.5b
(expm1 (/ 2 (+ (exp x) (exp (- x)))))
0.0b
(/ 2 (+ (exp x) (exp (- x))))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
2 → 29
Calls

2 calls. Slowest were:

6.0ms
(/ 2 (+ (exp x) (exp (- x))))
1.0ms
(expm1 (/ 2 (+ (exp x) (exp (- x)))))

series203.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

150.0ms
(expm1 (/ 2 (+ (exp x) (exp (- x)))))
53.0ms
(/ 2 (+ (exp x) (exp (- x))))

simplify751.0ms

Counts
13 → 35
Calls

13 calls. Slowest were:

368.0ms
(- (+ (* 1/3 (* (pow x 4) E)) E) (+ (* 1/2 (* (pow x 2) E)) 1))
250.0ms
(+ (* (exp x) (exp x)) (- (* (exp (- x)) (exp (- x))) (* (exp x) (exp (- x)))))
78.0ms
(- (+ (* 5/24 (pow x 4)) 1) (* 1/2 (pow x 2)))

prune453.0ms

Pruning

1 alts after pruning (0 fresh and 1 done)

Merged error: 0b