Hyperbolic tangent

Average Error: 58.1 → 0.0

Time: 6.9s

Precision: binary64

Cost: 19264

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]

Derivation

1. Initial program 58.1

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

2. Applied egg-rr 0.0

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

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	6464

\[\tanh x \]

Alternative 2
Error	1.8
Cost	576

\[\frac{1}{x \cdot 0.3333333333333333 + \frac{1}{x}} \]

Alternative 3
Error	2.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023012 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))