Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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. Final simplification 100.0%

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

Alternative 2: 80.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2}{4 - {x}^{4}} \cdot \left(2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 360.0) (* (/ 2.0 (- 4.0 (pow x 4.0))) (- 2.0 (* x x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = (2.0 / (4.0 - pow(x, 4.0))) * (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 / (4.0d0 - (x ** 4.0d0))) * (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 / (4.0 - Math.pow(x, 4.0))) * (2.0 - (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 360.0:
		tmp = (2.0 / (4.0 - math.pow(x, 4.0))) * (2.0 - (x * x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(Float64(2.0 / Float64(4.0 - (x ^ 4.0))) * 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 / (4.0 - (x ^ 4.0))) * (2.0 - (x * x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], N[(N[(2.0 / N[(4.0 - N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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. Taylor expanded in x around 0 82.8%

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

      1. unpow2 82.8%

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

   4. Simplified 82.8%

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

      1. flip-+ 75.6%

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

      2. associate-/r/ 75.6%

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

      3. metadata-eval 75.6%

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

      4. pow2 75.6%

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

      5. pow2 75.6%

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

      6. pow-prod-up 75.6%

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

      7. metadata-eval 75.6%

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

   6. Applied egg-rr 75.6%

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

   if 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 98.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2}{4 - {x}^{4}} \cdot \left(2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 74.7% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.45) (+ 1.0 (* (* x x) -0.5)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 67.8%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 67.8%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   4. Simplified 67.8%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]

   if 1.44999999999999996 < x

   1. Initial program 99.9%

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

   3. Applied egg-rr 95.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 87.4% accurate, 22.7× 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. Taylor expanded in x around 0 82.8%

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

      1. unpow2 82.8%

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

   4. Simplified 82.8%

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

   if 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 98.4%

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

4. Final simplification 86.4%

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

Alternative 5: 34.4% accurate, 67.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 350.0) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 350.0) {
		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 <= 350.0d0) 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 <= 350.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 350.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 350.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 350.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 350.0], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 350

   1. Initial program 100.0%

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

   3. Applied egg-rr 13.8%

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

   if 350 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 98.4%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 74.8% accurate, 67.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 350.0) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 350.0) {
		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 <= 350.0d0) 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 <= 350.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 350.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 350.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 350.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 350.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 350

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 67.0%

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

   if 350 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 98.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 50.8% 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%

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

3. Applied egg-rr 48.6%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 48.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023285 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))