Hyperbolic tangent

Percentage Accurate: 9.2% → 97.7%

Time: 5.6s

Alternatives: 5

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: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq 0.00092:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - t_0}{e^{x} + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x 0.00092)
     (/ (+ (* x 2.0) (* 0.3333333333333333 (pow x 3.0))) (+ 2.0 (* x x)))
     (/ (- (exp x) t_0) (+ (exp x) t_0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= 0.00092) {
		tmp = ((x * 2.0) + (0.3333333333333333 * pow(x, 3.0))) / (2.0 + (x * x));
	} else {
		tmp = (exp(x) - t_0) / (exp(x) + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= 0.00092d0) then
        tmp = ((x * 2.0d0) + (0.3333333333333333d0 * (x ** 3.0d0))) / (2.0d0 + (x * x))
    else
        tmp = (exp(x) - t_0) / (exp(x) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= 0.00092) {
		tmp = ((x * 2.0) + (0.3333333333333333 * Math.pow(x, 3.0))) / (2.0 + (x * x));
	} else {
		tmp = (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= 0.00092:
		tmp = ((x * 2.0) + (0.3333333333333333 * math.pow(x, 3.0))) / (2.0 + (x * x))
	else:
		tmp = (math.exp(x) - t_0) / (math.exp(x) + t_0)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= 0.00092)
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * (x ^ 3.0))) / Float64(2.0 + Float64(x * x)));
	else
		tmp = Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= 0.00092)
		tmp = ((x * 2.0) + (0.3333333333333333 * (x ^ 3.0))) / (2.0 + (x * x));
	else
		tmp = (exp(x) - t_0) / (exp(x) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, 0.00092], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.2000000000000003e-4

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 7.8%

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

      1. unpow2 7.8%

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

   4. Simplified 7.8%

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

   5. Taylor expanded in x around 0 98.3%

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

   if 9.2000000000000003e-4 < x

   1. Initial program 74.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00092:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot {x}^{3}}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\\ \end{array} \]

Alternative 2: 97.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x * 2.0), $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 10.1%

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

2. Taylor expanded in x around 0 7.8%

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

   1. unpow2 7.8%

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

4. Simplified 7.8%

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

5. Taylor expanded in x around 0 95.7%

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

6. Final simplification 95.7%

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

Alternative 3: 96.8% 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 10.1%

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

2. Taylor expanded in x around 0 95.2%

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

   1. count-2 95.2%

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

4. Simplified 95.2%

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

5. Taylor expanded in x around 0 95.5%

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

   1. unpow2 7.8%

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

7. Simplified 95.5%

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

8. Final simplification 95.5%

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

Alternative 4: 4.0% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 2.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.25)

C

double code(double x) {
	return 2.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.25d0
end function

Java

public static double code(double x) {
	return 2.25;
}

Python

def code(x):
	return 2.25

Julia

function code(x)
	return 2.25
end

MATLAB

function tmp = code(x)
	tmp = 2.25;
end

Wolfram

code[x_] := 2.25

Derivation

1. Initial program 10.1%

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

2. Taylor expanded in x around 0 95.5%

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

3. Taylor expanded in x around inf 5.6%

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

5. Applied egg-rr 4.0%

   \[\leadsto \color{blue}{2.25} + 0.13333333333333333 \cdot {x}^{5} \]

6. Taylor expanded in x around 0 4.1%

   \[\leadsto \color{blue}{2.25} \]

7. Final simplification 4.1%

   \[\leadsto 2.25 \]

Alternative 5: 96.8% 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 10.1%

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

2. Taylor expanded in x around 0 95.5%

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

3. Final simplification 95.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))