Hyperbolic tangent

Average Error: 58.2 → 1.1

Time: 3.2s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 1.90330772110840095e-6

   1. Initial program 58.9

      \[\]

   2. Taylor expanded around 0 1.1

      \[\leadsto \]

   3. Simplified 1.1

      \[\leadsto \]

   if 1.90330772110840095e-6 < x

   1. Initial program 30.1

      \[\]
   2. Using strategy rm

   3. Applied tanh-undef 0.1

      \[\leadsto \]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.7

      \[\leadsto \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (neg x))) (+ (exp x) (exp (neg x)))))