Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + \left(1 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (- 1.0 x))))

C

double code(double x) {
	return 2.0 / (exp(x) + (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + (1.0d0 - x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + (1.0 - x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + (1.0 - x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + (1.0 - x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 73.8%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 + -1 \cdot x\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg 73.8%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 73.8%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 73.8%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]
6. Add Preprocessing

Alternative 3: 94.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 370:\\ \;\;\;\;\frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 370.0)
   (/ 2.0 (+ 2.0 (* (* x x) (+ 1.0 (* (* x x) 0.08333333333333333)))))
   0.0))

C

double code(double x) {
	double tmp;
	if (x <= 370.0) {
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 370.0d0) then
        tmp = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + ((x * x) * 0.08333333333333333d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 370.0) {
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 370.0:
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 370.0)
		tmp = Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * 0.08333333333333333)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 370.0)
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 370.0], N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 370

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.8%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 91.8%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right)} \]

   5. Simplified 91.8%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)}} \]
   6. Step-by-step derivation

      1. unpow2 91.8%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   7. Applied egg-rr 91.8%

      \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]
   8. Step-by-step derivation

      1. unpow2 91.8%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   9. Applied egg-rr 91.8%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)} \]

   if 370 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.1% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 360.0) (/ 2.0 (+ 2.0 (* x x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = 2.0d0 / (2.0d0 + (x * x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 360.0:
		tmp = 2.0 / (2.0 + (x * x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = 2.0 / (2.0 + (x * x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 91.8%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 91.8%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right)} \]

   5. Simplified 91.8%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)}} \]
   6. Step-by-step derivation

      1. unpow2 91.8%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   7. Applied egg-rr 91.8%

      \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]
   8. Step-by-step derivation

      1. unpow2 91.8%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   9. Applied egg-rr 91.8%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)} \]

   10. Taylor expanded in x around 0 81.9%

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

   if 360 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 360.0) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 360.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.4%

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

   if 360 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.9% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 360.0) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 360.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   4. Applied egg-rr 13.2%

      \[\leadsto \color{blue}{0.5} \]

   if 360 < x

   1. Initial program 100.0%

      \[\frac{2}{e^{x} + e^{-x}} \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.1% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

4. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))