Hyperbolic tangent

Percentage Accurate: 9.0% → 99.6%

Time: 5.9s

Alternatives: 8

Speedup: 133.6×

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: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.3333333333333333 + \left({x}^{7} \cdot -0.05396825396825397 + {x}^{5} \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   -1.0
   (if (<= x 1.3)
     (+
      x
      (+
       (* (pow x 3.0) -0.3333333333333333)
       (+
        (* (pow x 7.0) -0.05396825396825397)
        (* (pow x 5.0) 0.13333333333333333))))
     1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = -1.0;
	} else if (x <= 1.3) {
		tmp = x + ((pow(x, 3.0) * -0.3333333333333333) + ((pow(x, 7.0) * -0.05396825396825397) + (pow(x, 5.0) * 0.13333333333333333)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = -1.0d0
    else if (x <= 1.3d0) then
        tmp = x + (((x ** 3.0d0) * (-0.3333333333333333d0)) + (((x ** 7.0d0) * (-0.05396825396825397d0)) + ((x ** 5.0d0) * 0.13333333333333333d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = -1.0;
	} else if (x <= 1.3) {
		tmp = x + ((Math.pow(x, 3.0) * -0.3333333333333333) + ((Math.pow(x, 7.0) * -0.05396825396825397) + (Math.pow(x, 5.0) * 0.13333333333333333)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = -1.0
	elif x <= 1.3:
		tmp = x + ((math.pow(x, 3.0) * -0.3333333333333333) + ((math.pow(x, 7.0) * -0.05396825396825397) + (math.pow(x, 5.0) * 0.13333333333333333)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = -1.0;
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(Float64((x ^ 3.0) * -0.3333333333333333) + Float64(Float64((x ^ 7.0) * -0.05396825396825397) + Float64((x ^ 5.0) * 0.13333333333333333))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = -1.0;
	elseif (x <= 1.3)
		tmp = x + (((x ^ 3.0) * -0.3333333333333333) + (((x ^ 7.0) * -0.05396825396825397) + ((x ^ 5.0) * 0.13333333333333333)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], -1.0, If[LessEqual[x, 1.3], N[(x + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[(N[Power[x, 7.0], $MachinePrecision] * -0.05396825396825397), $MachinePrecision] + N[(N[Power[x, 5.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 39.7%

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

   2. Taylor expanded in x around 0 7.4%

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

   3. Taylor expanded in x around 0 6.6%

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

      1. *-commutative 6.6%

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

   5. Simplified 6.6%

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

   7. Applied egg-rr 84.9%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 11.1%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 1.30000000000000004 < x

   1. Initial program 24.8%

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

   2. Taylor expanded in x around 0 6.6%

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

   3. Taylor expanded in x around 0 6.1%

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

      1. *-commutative 6.1%

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

   5. Simplified 6.1%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.3333333333333333 + \left({x}^{7} \cdot -0.05396825396825397 + {x}^{5} \cdot 0.13333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.75:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)\right)}{2 + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.75)
   (/
    (+
     (* 0.0003968253968253968 (pow x 7.0))
     (+
      (* 0.016666666666666666 (pow x 5.0))
      (+ (* 0.3333333333333333 (pow x 3.0)) (* x 2.0))))
    (+
     2.0
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (+ (* 0.08333333333333333 (pow x 4.0)) (pow x 2.0)))))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 2.75) {
		tmp = ((0.0003968253968253968 * pow(x, 7.0)) + ((0.016666666666666666 * pow(x, 5.0)) + ((0.3333333333333333 * pow(x, 3.0)) + (x * 2.0)))) / (2.0 + ((0.002777777777777778 * pow(x, 6.0)) + ((0.08333333333333333 * pow(x, 4.0)) + pow(x, 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.75d0) then
        tmp = ((0.0003968253968253968d0 * (x ** 7.0d0)) + ((0.016666666666666666d0 * (x ** 5.0d0)) + ((0.3333333333333333d0 * (x ** 3.0d0)) + (x * 2.0d0)))) / (2.0d0 + ((0.002777777777777778d0 * (x ** 6.0d0)) + ((0.08333333333333333d0 * (x ** 4.0d0)) + (x ** 2.0d0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.75) {
		tmp = ((0.0003968253968253968 * Math.pow(x, 7.0)) + ((0.016666666666666666 * Math.pow(x, 5.0)) + ((0.3333333333333333 * Math.pow(x, 3.0)) + (x * 2.0)))) / (2.0 + ((0.002777777777777778 * Math.pow(x, 6.0)) + ((0.08333333333333333 * Math.pow(x, 4.0)) + Math.pow(x, 2.0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.75:
		tmp = ((0.0003968253968253968 * math.pow(x, 7.0)) + ((0.016666666666666666 * math.pow(x, 5.0)) + ((0.3333333333333333 * math.pow(x, 3.0)) + (x * 2.0)))) / (2.0 + ((0.002777777777777778 * math.pow(x, 6.0)) + ((0.08333333333333333 * math.pow(x, 4.0)) + math.pow(x, 2.0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.75)
		tmp = Float64(Float64(Float64(0.0003968253968253968 * (x ^ 7.0)) + Float64(Float64(0.016666666666666666 * (x ^ 5.0)) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(x * 2.0)))) / Float64(2.0 + Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + (x ^ 2.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.75)
		tmp = ((0.0003968253968253968 * (x ^ 7.0)) + ((0.016666666666666666 * (x ^ 5.0)) + ((0.3333333333333333 * (x ^ 3.0)) + (x * 2.0)))) / (2.0 + ((0.002777777777777778 * (x ^ 6.0)) + ((0.08333333333333333 * (x ^ 4.0)) + (x ^ 2.0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.75], N[(N[(N[(0.0003968253968253968 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.75

   1. Initial program 12.0%

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

   2. Taylor expanded in x around 0 97.7%

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

   3. Taylor expanded in x around 0 97.8%

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

   if 2.75 < x

   1. Initial program 14.1%

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

   2. Taylor expanded in x around 0 4.6%

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

   3. Taylor expanded in x around 0 4.4%

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

      1. *-commutative 4.4%

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

   5. Simplified 4.4%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.75:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)\right)}{2 + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)\right)}{e^{x} + e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.0)
   (/
    (+
     (* 0.0003968253968253968 (pow x 7.0))
     (+
      (* 0.016666666666666666 (pow x 5.0))
      (+ (* 0.3333333333333333 (pow x 3.0)) (* x 2.0))))
    (+ (exp x) (exp (- x))))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = ((0.0003968253968253968 * pow(x, 7.0)) + ((0.016666666666666666 * pow(x, 5.0)) + ((0.3333333333333333 * pow(x, 3.0)) + (x * 2.0)))) / (exp(x) + exp(-x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.0d0) then
        tmp = ((0.0003968253968253968d0 * (x ** 7.0d0)) + ((0.016666666666666666d0 * (x ** 5.0d0)) + ((0.3333333333333333d0 * (x ** 3.0d0)) + (x * 2.0d0)))) / (exp(x) + exp(-x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = ((0.0003968253968253968 * Math.pow(x, 7.0)) + ((0.016666666666666666 * Math.pow(x, 5.0)) + ((0.3333333333333333 * Math.pow(x, 3.0)) + (x * 2.0)))) / (Math.exp(x) + Math.exp(-x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.0:
		tmp = ((0.0003968253968253968 * math.pow(x, 7.0)) + ((0.016666666666666666 * math.pow(x, 5.0)) + ((0.3333333333333333 * math.pow(x, 3.0)) + (x * 2.0)))) / (math.exp(x) + math.exp(-x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(Float64(Float64(0.0003968253968253968 * (x ^ 7.0)) + Float64(Float64(0.016666666666666666 * (x ^ 5.0)) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(x * 2.0)))) / Float64(exp(x) + exp(Float64(-x))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.0)
		tmp = ((0.0003968253968253968 * (x ^ 7.0)) + ((0.016666666666666666 * (x ^ 5.0)) + ((0.3333333333333333 * (x ^ 3.0)) + (x * 2.0)))) / (exp(x) + exp(-x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.0], N[(N[(N[(0.0003968253968253968 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 12.0%

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

   2. Taylor expanded in x around 0 97.7%

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

   if 3 < x

   1. Initial program 14.1%

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

   2. Taylor expanded in x around 0 4.6%

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

   3. Taylor expanded in x around 0 4.4%

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

      1. *-commutative 4.4%

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

   5. Simplified 4.4%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{0.0003968253968253968 \cdot {x}^{7} + \left(0.016666666666666666 \cdot {x}^{5} + \left(0.3333333333333333 \cdot {x}^{3} + x \cdot 2\right)\right)}{e^{x} + e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.3333333333333333 + {x}^{5} \cdot 0.13333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25)
   (+
    x
    (+
     (* (pow x 3.0) -0.3333333333333333)
     (* (pow x 5.0) 0.13333333333333333)))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x + ((pow(x, 3.0) * -0.3333333333333333) + (pow(x, 5.0) * 0.13333333333333333));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = x + (((x ** 3.0d0) * (-0.3333333333333333d0)) + ((x ** 5.0d0) * 0.13333333333333333d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x + ((Math.pow(x, 3.0) * -0.3333333333333333) + (Math.pow(x, 5.0) * 0.13333333333333333));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = x + ((math.pow(x, 3.0) * -0.3333333333333333) + (math.pow(x, 5.0) * 0.13333333333333333))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(x + Float64(Float64((x ^ 3.0) * -0.3333333333333333) + Float64((x ^ 5.0) * 0.13333333333333333)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = x + (((x ^ 3.0) * -0.3333333333333333) + ((x ^ 5.0) * 0.13333333333333333));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(x + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + N[(N[Power[x, 5.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 11.6%

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

   2. Taylor expanded in x around 0 97.6%

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

   if 1.25 < x

   1. Initial program 24.8%

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

   2. Taylor expanded in x around 0 6.6%

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

   3. Taylor expanded in x around 0 6.1%

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

      1. *-commutative 6.1%

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

   5. Simplified 6.1%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.3333333333333333 + {x}^{5} \cdot 0.13333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 99.3% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   -1.0
   (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 = -1.0;
	} else 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 = -1.0d0
    else 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 = -1.0;
	} else 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 = -1.0
	elif 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 = -1.0;
	elseif (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 = -1.0;
	elseif (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], -1.0, 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 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 39.7%

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

   2. Taylor expanded in x around 0 7.4%

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

   3. Taylor expanded in x around 0 6.6%

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

      1. *-commutative 6.6%

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

   5. Simplified 6.6%

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

   7. Applied egg-rr 84.9%

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

   if -1.1499999999999999 < x < 1.1499999999999999

   1. Initial program 11.1%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   if 1.1499999999999999 < x

   1. Initial program 24.8%

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

   2. Taylor expanded in x around 0 6.6%

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

   3. Taylor expanded in x around 0 6.1%

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

      1. *-commutative 6.1%

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

   5. Simplified 6.1%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 98.4%

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

Alternative 6: 7.7% accurate, 133.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1e-309) -1.0 1.0))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 13.3%

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

   2. Taylor expanded in x around 0 95.1%

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

   3. Taylor expanded in x around 0 93.7%

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

      1. *-commutative 93.7%

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

   5. Simplified 93.7%

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

   7. Applied egg-rr 8.7%

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

   if -1.000000000000002e-309 < x

   1. Initial program 10.8%

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

   2. Taylor expanded in x around 0 93.8%

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

   3. Taylor expanded in x around 0 92.8%

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

      1. *-commutative 92.8%

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

   5. Simplified 92.8%

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

   7. Applied egg-rr 10.7%

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

4. Final simplification 9.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 97.8% accurate, 133.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) x 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], x, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 11.6%

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

   2. Taylor expanded in x around 0 96.1%

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

   if 1 < x

   1. Initial program 24.8%

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

   2. Taylor expanded in x around 0 6.6%

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

   3. Taylor expanded in x around 0 6.1%

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

      1. *-commutative 6.1%

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

   5. Simplified 6.1%

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 5.1% 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 12.1%

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

2. Taylor expanded in x around 0 94.4%

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

3. Taylor expanded in x around 0 93.2%

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

   1. *-commutative 93.2%

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

5. Simplified 93.2%

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

7. Applied egg-rr 5.5%

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

8. Final simplification 5.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023309 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))