Average Error: 58.1 → 1.7
Time: 5.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]
↓
\[\mathsf{fma}\left({x}^{5}, \frac{2}{15}, x - \frac{1}{3} \cdot {x}^{3}\right)\]
\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}
↓
\mathsf{fma}\left({x}^{5}, \frac{2}{15}, x - \frac{1}{3} \cdot {x}^{3}\right)
double code(double x) {
	return ((double) (((double) (((double) exp(x)) - ((double) exp(((double) -(x)))))) / ((double) (((double) exp(x)) + ((double) exp(((double) -(x))))))));
}
↓
double code(double x) {
	return ((double) fma(((double) pow(x, 5.0)), 0.13333333333333333, ((double) (x - ((double) (0.3333333333333333 * ((double) pow(x, 3.0))))))));
}

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

    \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\]

  2. Simplified0.7

    \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{fma}\left(1, 1, e^{x + x}\right)}}\]

  3. Taylor expanded around 0 1.7

    \[\leadsto 1 \cdot \color{blue}{\left(\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}\right)}\]

  4. Simplified1.7

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left({x}^{5}, \frac{2}{15}, x - \frac{1}{3} \cdot {x}^{3}\right)}\]

  5. Final simplification1.7

    \[\leadsto \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x - \frac{1}{3} \cdot {x}^{3}\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))