Hyperbolic tangent

Average Error: 58.2 → 0.0

Time: 3.9s

Precision: binary64

Cost: 6464

Math

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

↓

\[\tanh x \]

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / (exp(x) + exp(-x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = tanh(x)
end function

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.tanh(x);
}

Python

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

↓

def code(x):
	return 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 tanh(x)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / (exp(x) + exp(-x));
end

↓

function tmp = code(x)
	tmp = 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[Tanh[x], $MachinePrecision]

Derivation

1. Initial program 58.2

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

2. Taylor expanded in x around inf 58.2

   \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{e^{-x} + e^{x}}} \]

3. Simplified 0.0

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

   [Start] 58.2	\[ \frac{e^{x} - e^{-x}}{e^{-x} + e^{x}} \]
   +-commutative [<=] 58.2	\[ \frac{e^{x} - e^{-x}}{\color{blue}{e^{x} + e^{-x}}} \]
   tanh-def-a [<=] 0.0	\[ \color{blue}{\tanh x} \]

4. Final simplification 0.0

   \[\leadsto \tanh x \]

Alternatives

Alternative 1
Error	2.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023040 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))