Hyperbolic tangent

Percentage Accurate: 9.0% → 98.1%

Time: 5.2s

Alternatives: 6

Speedup: 409.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t_0}{e^{x} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 9.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t_0}{e^{x} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{-x_m}\\ t_1 := \frac{e^{x_m} - t_0}{e^{x_m} + t_0}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0.005:\\ \;\;\;\;x_m + \left({x_m}^{3} \cdot -0.3333333333333333 + \left({x_m}^{7} \cdot -0.05396825396825397 + {x_m}^{5} \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))) (t_1 (/ (- (exp x_m) t_0) (+ (exp x_m) t_0))))
   (*
    x_s
    (if (<= t_1 0.005)
      (+
       x_m
       (+
        (* (pow x_m 3.0) -0.3333333333333333)
        (+
         (* (pow x_m 7.0) -0.05396825396825397)
         (* (pow x_m 5.0) 0.13333333333333333))))
      t_1))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = exp(-x_m);
	double t_1 = (exp(x_m) - t_0) / (exp(x_m) + t_0);
	double tmp;
	if (t_1 <= 0.005) {
		tmp = x_m + ((pow(x_m, 3.0) * -0.3333333333333333) + ((pow(x_m, 7.0) * -0.05396825396825397) + (pow(x_m, 5.0) * 0.13333333333333333)));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-x_m)
    t_1 = (exp(x_m) - t_0) / (exp(x_m) + t_0)
    if (t_1 <= 0.005d0) then
        tmp = x_m + (((x_m ** 3.0d0) * (-0.3333333333333333d0)) + (((x_m ** 7.0d0) * (-0.05396825396825397d0)) + ((x_m ** 5.0d0) * 0.13333333333333333d0)))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = Math.exp(-x_m);
	double t_1 = (Math.exp(x_m) - t_0) / (Math.exp(x_m) + t_0);
	double tmp;
	if (t_1 <= 0.005) {
		tmp = x_m + ((Math.pow(x_m, 3.0) * -0.3333333333333333) + ((Math.pow(x_m, 7.0) * -0.05396825396825397) + (Math.pow(x_m, 5.0) * 0.13333333333333333)));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = math.exp(-x_m)
	t_1 = (math.exp(x_m) - t_0) / (math.exp(x_m) + t_0)
	tmp = 0
	if t_1 <= 0.005:
		tmp = x_m + ((math.pow(x_m, 3.0) * -0.3333333333333333) + ((math.pow(x_m, 7.0) * -0.05396825396825397) + (math.pow(x_m, 5.0) * 0.13333333333333333)))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = exp(Float64(-x_m))
	t_1 = Float64(Float64(exp(x_m) - t_0) / Float64(exp(x_m) + t_0))
	tmp = 0.0
	if (t_1 <= 0.005)
		tmp = Float64(x_m + Float64(Float64((x_m ^ 3.0) * -0.3333333333333333) + Float64(Float64((x_m ^ 7.0) * -0.05396825396825397) + Float64((x_m ^ 5.0) * 0.13333333333333333))));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = exp(-x_m);
	t_1 = (exp(x_m) - t_0) / (exp(x_m) + t_0);
	tmp = 0.0;
	if (t_1 <= 0.005)
		tmp = x_m + (((x_m ^ 3.0) * -0.3333333333333333) + (((x_m ^ 7.0) * -0.05396825396825397) + ((x_m ^ 5.0) * 0.13333333333333333)));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[x$95$m], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 0.005], N[(x$95$m + N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[(N[Power[x$95$m, 7.0], $MachinePrecision] * -0.05396825396825397), $MachinePrecision] + N[(N[Power[x$95$m, 5.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 0.0050000000000000001

   1. Initial program 10.8%

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

   2. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + \left(-0.05396825396825397 \cdot {x}^{7} + 0.13333333333333333 \cdot {x}^{5}\right)\right)} \]

   if 0.0050000000000000001 < (/.f64 (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 66.1%

      \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \leq 0.005:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.3333333333333333 + \left({x}^{7} \cdot -0.05396825396825397 + {x}^{5} \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \frac{0.0003968253968253968 \cdot {x_m}^{7} + \left(0.016666666666666666 \cdot {x_m}^{5} + \left(0.3333333333333333 \cdot {x_m}^{3} + x_m \cdot 2\right)\right)}{e^{x_m} + e^{-x_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (/
   (+
    (* 0.0003968253968253968 (pow x_m 7.0))
    (+
     (* 0.016666666666666666 (pow x_m 5.0))
     (+ (* 0.3333333333333333 (pow x_m 3.0)) (* x_m 2.0))))
   (+ (exp x_m) (exp (- x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (((0.0003968253968253968 * pow(x_m, 7.0)) + ((0.016666666666666666 * pow(x_m, 5.0)) + ((0.3333333333333333 * pow(x_m, 3.0)) + (x_m * 2.0)))) / (exp(x_m) + exp(-x_m)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (((0.0003968253968253968d0 * (x_m ** 7.0d0)) + ((0.016666666666666666d0 * (x_m ** 5.0d0)) + ((0.3333333333333333d0 * (x_m ** 3.0d0)) + (x_m * 2.0d0)))) / (exp(x_m) + exp(-x_m)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (((0.0003968253968253968 * Math.pow(x_m, 7.0)) + ((0.016666666666666666 * Math.pow(x_m, 5.0)) + ((0.3333333333333333 * Math.pow(x_m, 3.0)) + (x_m * 2.0)))) / (Math.exp(x_m) + Math.exp(-x_m)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (((0.0003968253968253968 * math.pow(x_m, 7.0)) + ((0.016666666666666666 * math.pow(x_m, 5.0)) + ((0.3333333333333333 * math.pow(x_m, 3.0)) + (x_m * 2.0)))) / (math.exp(x_m) + math.exp(-x_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(Float64(0.0003968253968253968 * (x_m ^ 7.0)) + Float64(Float64(0.016666666666666666 * (x_m ^ 5.0)) + Float64(Float64(0.3333333333333333 * (x_m ^ 3.0)) + Float64(x_m * 2.0)))) / Float64(exp(x_m) + exp(Float64(-x_m)))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (((0.0003968253968253968 * (x_m ^ 7.0)) + ((0.016666666666666666 * (x_m ^ 5.0)) + ((0.3333333333333333 * (x_m ^ 3.0)) + (x_m * 2.0)))) / (exp(x_m) + exp(-x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(N[(0.0003968253968253968 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x$95$m], $MachinePrecision] + N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.7%

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

2. Taylor expanded in x around 0 97.4%

   \[\leadsto \frac{\color{blue}{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + 2 \cdot x\right)\right)}}{e^{x} + e^{-x}} \]

3. Final simplification 97.4%

   \[\leadsto \frac{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)\right)}{e^{x} + e^{-x}} \]

Alternative 3: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + \left({x_m}^{3} \cdot -0.3333333333333333 + {x_m}^{5} \cdot 0.13333333333333333\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (+
   x_m
   (+
    (* (pow x_m 3.0) -0.3333333333333333)
    (* (pow x_m 5.0) 0.13333333333333333)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m + ((pow(x_m, 3.0) * -0.3333333333333333) + (pow(x_m, 5.0) * 0.13333333333333333)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m + (((x_m ** 3.0d0) * (-0.3333333333333333d0)) + ((x_m ** 5.0d0) * 0.13333333333333333d0)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m + ((Math.pow(x_m, 3.0) * -0.3333333333333333) + (Math.pow(x_m, 5.0) * 0.13333333333333333)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m + ((math.pow(x_m, 3.0) * -0.3333333333333333) + (math.pow(x_m, 5.0) * 0.13333333333333333)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m + Float64(Float64((x_m ^ 3.0) * -0.3333333333333333) + Float64((x_m ^ 5.0) * 0.13333333333333333))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m + (((x_m ^ 3.0) * -0.3333333333333333) + ((x_m ^ 5.0) * 0.13333333333333333)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m + N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[Power[x$95$m, 5.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.7%

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

2. Taylor expanded in x around 0 96.9%

   \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

3. Final simplification 96.9%

   \[\leadsto x + \left({x}^{3} \cdot -0.3333333333333333 + {x}^{5} \cdot 0.13333333333333333\right) \]

Alternative 4: 96.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \left(x_m + {x_m}^{3} \cdot -0.3333333333333333\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (+ x_m (* (pow x_m 3.0) -0.3333333333333333))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m + (pow(x_m, 3.0) * -0.3333333333333333));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m + ((x_m ** 3.0d0) * (-0.3333333333333333d0)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m + (Math.pow(x_m, 3.0) * -0.3333333333333333));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m + (math.pow(x_m, 3.0) * -0.3333333333333333))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m + Float64((x_m ^ 3.0) * -0.3333333333333333)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m + ((x_m ^ 3.0) * -0.3333333333333333));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.7%

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

2. Taylor expanded in x around 0 96.3%

   \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
3. Step-by-step derivation

   1. *-commutative 96.3%

      \[\leadsto x + \color{blue}{{x}^{3} \cdot -0.3333333333333333} \]

4. Simplified 96.3%

   \[\leadsto \color{blue}{x + {x}^{3} \cdot -0.3333333333333333} \]

5. Final simplification 96.3%

   \[\leadsto x + {x}^{3} \cdot -0.3333333333333333 \]

Alternative 5: 5.7% accurate, 409.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot 2.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s 2.5))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 2.5;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * 2.5d0
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * 2.5;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 2.5

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 2.5)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 2.5;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * 2.5), $MachinePrecision]

Derivation

1. Initial program 12.7%

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

3. Applied egg-rr 4.0%

   \[\leadsto \frac{\color{blue}{5}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around 0 4.0%

   \[\leadsto \color{blue}{2.5} \]

5. Final simplification 4.0%

   \[\leadsto 2.5 \]

Alternative 6: 96.7% accurate, 409.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot x_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 12.7%

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

2. Taylor expanded in x around 0 95.3%

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

3. Final simplification 95.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023338 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))