Hyperbolic tangent

Average Error: 58.3 → 0.0

Time: 3.5s

Precision: binary64

Math

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

↓

\[\mathsf{log1p}\left(\mathsf{expm1}\left(\tanh x\right)\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))

↓

(FPCore (x) :precision binary64 (log1p (expm1 (tanh x))))

C

double code(double x) {
	return (exp(x) - exp(-x)) / (exp(x) + exp(-x));
}

↓

double code(double x) {
	return log1p(expm1(tanh(x)));
}

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / (Math.exp(x) + Math.exp(-x));
}

↓

public static double code(double x) {
	return Math.log1p(Math.expm1(Math.tanh(x)));
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / (math.exp(x) + math.exp(-x))

↓

def code(x):
	return math.log1p(math.expm1(math.tanh(x)))

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / Float64(exp(x) + exp(Float64(-x))))
end

↓

function code(x)
	return log1p(expm1(tanh(x)))
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[Log[1 + N[(Exp[N[Tanh[x], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Error

Bits error versus x

Derivation

1. Initial program 58.3

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

2. Applied tanh-undef_binary64 0.0

   \[\leadsto \color{blue}{\tanh x} \]

3. Applied log1p-expm1-u_binary64 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tanh x\right)\right)} \]

4. Final simplification 0.0

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\tanh x\right)\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))