Hyperbolic tangent

Percentage Accurate: 9.2% → 98.0%

Time: 6.7s

Alternatives: 10

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: 9.2% 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: 98.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x + {x}^{3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.15) (+ x (* (pow x 3.0) -0.3333333333333333)) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = x + (pow(x, 3.0) * -0.3333333333333333);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.15d0) then
        tmp = x + ((x ** 3.0d0) * (-0.3333333333333333d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = x + (Math.pow(x, 3.0) * -0.3333333333333333);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.15:
		tmp = x + (math.pow(x, 3.0) * -0.3333333333333333)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.15)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.3333333333333333));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.15)
		tmp = x + ((x ^ 3.0) * -0.3333333333333333);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.15], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0 99.0%

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

   if 1.1499999999999999 < x

   1. Initial program 33.1%

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

   3. Applied egg-rr 8.1%

      \[\leadsto \frac{\color{blue}{27}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 12.6%

      \[\leadsto \frac{27}{\color{blue}{2 + {x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 12.6%

         \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

   6. Simplified 12.6%

      \[\leadsto \frac{27}{\color{blue}{2 + x \cdot x}} \]

   8. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;x + {x}^{3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2 + \mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (* 2.0 x)
   (+
    (* 0.3333333333333333 (pow x 3.0))
    (+
     (* 0.0003968253968253968 (pow x 7.0))
     (* 0.016666666666666666 (pow x 5.0)))))
  (+
   2.0
   (fma
    0.002777777777777778
    (pow x 6.0)
    (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))))))

C

double code(double x) {
	return ((2.0 * x) + ((0.3333333333333333 * pow(x, 3.0)) + ((0.0003968253968253968 * pow(x, 7.0)) + (0.016666666666666666 * pow(x, 5.0))))) / (2.0 + fma(0.002777777777777778, pow(x, 6.0), ((x * x) + (0.08333333333333333 * pow(x, 4.0)))));
}

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(0.0003968253968253968 * (x ^ 7.0)) + Float64(0.016666666666666666 * (x ^ 5.0))))) / Float64(2.0 + fma(0.002777777777777778, (x ^ 6.0), Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0))))))
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0003968253968253968 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 97.3%

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

3. Taylor expanded in x around 0 97.4%

   \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{\color{blue}{2 + \left(0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)\right)}} \]
4. Step-by-step derivation

   1. fma-def 97.4%

      \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2 + \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

   2. unpow2 97.4%

      \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2 + \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

5. Simplified 97.4%

   \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{\color{blue}{2 + \mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

6. Final simplification 97.4%

   \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2 + \mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

Alternative 3: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (* 2.0 x)
   (+ (* 0.3333333333333333 (pow x 3.0)) (* 0.016666666666666666 (pow x 5.0))))
  (+ 2.0 (+ (* x x) (* 0.08333333333333333 (pow x 4.0))))))

C

double code(double x) {
	return ((2.0 * x) + ((0.3333333333333333 * pow(x, 3.0)) + (0.016666666666666666 * pow(x, 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * pow(x, 4.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 * x) + ((0.3333333333333333d0 * (x ** 3.0d0)) + (0.016666666666666666d0 * (x ** 5.0d0)))) / (2.0d0 + ((x * x) + (0.08333333333333333d0 * (x ** 4.0d0))))
end function

Java

public static double code(double x) {
	return ((2.0 * x) + ((0.3333333333333333 * Math.pow(x, 3.0)) + (0.016666666666666666 * Math.pow(x, 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * Math.pow(x, 4.0))));
}

Python

def code(x):
	return ((2.0 * x) + ((0.3333333333333333 * math.pow(x, 3.0)) + (0.016666666666666666 * math.pow(x, 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * math.pow(x, 4.0))))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(0.016666666666666666 * (x ^ 5.0)))) / Float64(2.0 + Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 * x) + ((0.3333333333333333 * (x ^ 3.0)) + (0.016666666666666666 * (x ^ 5.0)))) / (2.0 + ((x * x) + (0.08333333333333333 * (x ^ 4.0))));
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 8.4%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
3. Step-by-step derivation

   1. unpow2 8.4%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 8.4%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

5. Taylor expanded in x around 0 97.3%

   \[\leadsto \frac{\color{blue}{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

6. Final simplification 97.3%

   \[\leadsto \frac{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

Alternative 4: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+ (* 2.0 x) (* 0.3333333333333333 (pow x 3.0)))
  (+ 2.0 (+ (* x x) (* 0.08333333333333333 (pow x 4.0))))))

C

double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * pow(x, 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * pow(x, 4.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 * x) + (0.3333333333333333d0 * (x ** 3.0d0))) / (2.0d0 + ((x * x) + (0.08333333333333333d0 * (x ** 4.0d0))))
end function

Java

public static double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * Math.pow(x, 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * Math.pow(x, 4.0))));
}

Python

def code(x):
	return ((2.0 * x) + (0.3333333333333333 * math.pow(x, 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * math.pow(x, 4.0))))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(0.3333333333333333 * (x ^ 3.0))) / Float64(2.0 + Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 * x) + (0.3333333333333333 * (x ^ 3.0))) / (2.0 + ((x * x) + (0.08333333333333333 * (x ^ 4.0))));
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 8.4%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)}} \]
3. Step-by-step derivation

   1. unpow2 8.4%

      \[\leadsto \frac{e^{x} - e^{-x}}{2 + \left(\color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

4. Simplified 8.4%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}} \]

5. Taylor expanded in x around 0 97.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

6. Final simplification 97.2%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2 + \left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

Alternative 5: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2 + x \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma 2.0 x (* 0.3333333333333333 (pow x 3.0))) (/ 1.0 (+ 2.0 (* x x)))))

C

double code(double x) {
	return fma(2.0, x, (0.3333333333333333 * pow(x, 3.0))) * (1.0 / (2.0 + (x * x)));
}

Julia

function code(x)
	return Float64(fma(2.0, x, Float64(0.3333333333333333 * (x ^ 3.0))) * Float64(1.0 / Float64(2.0 + Float64(x * x))))
end

Wolfram

code[x_] := N[(N[(2.0 * x + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 8.3%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 4.0%

      \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

4. Simplified 8.3%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + x \cdot x}} \]

5. Taylor expanded in x around 0 97.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2 + x \cdot x} \]
6. Step-by-step derivation

   1. div-inv 97.2%

      \[\leadsto \color{blue}{\left(2 \cdot x + 0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2 + x \cdot x}} \]

   2. fma-def 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right)} \cdot \frac{1}{2 + x \cdot x} \]

7. Applied egg-rr 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2 + x \cdot x}} \]

8. Final simplification 97.2%

   \[\leadsto \mathsf{fma}\left(2, x, 0.3333333333333333 \cdot {x}^{3}\right) \cdot \frac{1}{2 + x \cdot x} \]

Alternative 6: 97.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2 + x \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* 2.0 x) (* 0.3333333333333333 (pow x 3.0))) (+ 2.0 (* x x))))

C

double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * pow(x, 3.0))) / (2.0 + (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.0d0 * x) + (0.3333333333333333d0 * (x ** 3.0d0))) / (2.0d0 + (x * x))
end function

Java

public static double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * Math.pow(x, 3.0))) / (2.0 + (x * x));
}

Python

def code(x):
	return ((2.0 * x) + (0.3333333333333333 * math.pow(x, 3.0))) / (2.0 + (x * x))

Julia

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(0.3333333333333333 * (x ^ 3.0))) / Float64(2.0 + Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = ((2.0 * x) + (0.3333333333333333 * (x ^ 3.0))) / (2.0 + (x * x));
end

Wolfram

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 8.3%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 4.0%

      \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

4. Simplified 8.3%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + x \cdot x}} \]

5. Taylor expanded in x around 0 97.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2 + x \cdot x} \]

6. Final simplification 97.2%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2 + x \cdot x} \]

Alternative 7: 96.6% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \frac{x + x}{2 + x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (+ x x) (+ 2.0 (* x x))))

C

double code(double x) {
	return (x + x) / (2.0 + (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + x) / (2.0d0 + (x * x))
end function

Java

public static double code(double x) {
	return (x + x) / (2.0 + (x * x));
}

Python

def code(x):
	return (x + x) / (2.0 + (x * x))

Julia

function code(x)
	return Float64(Float64(x + x) / Float64(2.0 + Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (x + x) / (2.0 + (x * x));
end

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 8.3%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + {x}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 4.0%

      \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

4. Simplified 8.3%

   \[\leadsto \frac{e^{x} - e^{-x}}{\color{blue}{2 + x \cdot x}} \]

5. Taylor expanded in x around 0 96.8%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2 + x \cdot x} \]
6. Step-by-step derivation

   1. count-2 96.8%

      \[\leadsto \frac{\color{blue}{x + x}}{2 + x \cdot x} \]

7. Simplified 96.8%

   \[\leadsto \frac{\color{blue}{x + x}}{2 + x \cdot x} \]

8. Final simplification 96.8%

   \[\leadsto \frac{x + x}{2 + x \cdot x} \]

Alternative 8: 7.7% accurate, 133.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5e-311) -1.0 1.0))

C

double code(double x) {
	double tmp;
	if (x <= -5e-311) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-311)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-311) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-311:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-311)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-311)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-311], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000023e-311

   1. Initial program 9.9%

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

   3. Applied egg-rr 2.1%

      \[\leadsto \frac{\color{blue}{27}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 2.1%

      \[\leadsto \frac{27}{\color{blue}{2 + {x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 2.1%

         \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

   6. Simplified 2.1%

      \[\leadsto \frac{27}{\color{blue}{2 + x \cdot x}} \]

   8. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{-1} \]

   if -5.00000000000023e-311 < x

   1. Initial program 8.7%

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

   3. Applied egg-rr 5.4%

      \[\leadsto \frac{\color{blue}{27}}{e^{x} + e^{-x}} \]

   4. Taylor expanded in x around 0 5.6%

      \[\leadsto \frac{27}{\color{blue}{2 + {x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 5.6%

         \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

   6. Simplified 5.6%

      \[\leadsto \frac{27}{\color{blue}{2 + x \cdot x}} \]

   8. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 5.0% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 9.3%

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

3. Applied egg-rr 3.9%

   \[\leadsto \frac{\color{blue}{27}}{e^{x} + e^{-x}} \]

4. Taylor expanded in x around 0 4.0%

   \[\leadsto \frac{27}{\color{blue}{2 + {x}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 4.0%

      \[\leadsto \frac{27}{2 + \color{blue}{x \cdot x}} \]

6. Simplified 4.0%

   \[\leadsto \frac{27}{\color{blue}{2 + x \cdot x}} \]

8. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{-1} \]

9. Final simplification 4.4%

   \[\leadsto -1 \]

Alternative 10: 96.6% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 9.3%

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

2. Taylor expanded in x around 0 96.7%

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

3. Final simplification 96.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023258 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))