Hyperbolic secant

Percentage Accurate: 100.0% → 99.4%

Time: 2.5s

Alternatives: 5

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: 99.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 340:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.20833333333333334\right) + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 340.0)
   (+ 1.0 (* (* x x) (+ (* x (* x 0.20833333333333334)) -0.5)))
   0.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 340.0) {
		tmp = 1.0 + ((x * x) * ((x * (x * 0.20833333333333334)) + -0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 340.0d0) then
        tmp = 1.0d0 + ((x * x) * ((x * (x * 0.20833333333333334d0)) + (-0.5d0)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 340.0) {
		tmp = 1.0 + ((x * x) * ((x * (x * 0.20833333333333334)) + -0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 340.0:
		tmp = 1.0 + ((x * x) * ((x * (x * 0.20833333333333334)) + -0.5))
	else:
		tmp = 0.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 340.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * Float64(x * 0.20833333333333334)) + -0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 340.0)
		tmp = 1.0 + ((x * x) * ((x * (x * 0.20833333333333334)) + -0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 340.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * 0.20833333333333334), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 340

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 65.1%

      \[\leadsto \color{blue}{1 + \left(-0.5 \cdot {x}^{2} + 0.20833333333333334 \cdot {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. fma-def 65.1%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-0.5, {x}^{2}, 0.20833333333333334 \cdot {x}^{4}\right)} \]

      2. unpow2 65.1%

         \[\leadsto 1 + \mathsf{fma}\left(-0.5, \color{blue}{x \cdot x}, 0.20833333333333334 \cdot {x}^{4}\right) \]

   4. Simplified 65.1%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(-0.5, x \cdot x, 0.20833333333333334 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 65.1%

         \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot \left(x \cdot x\right) + 0.20833333333333334 \cdot {x}^{4}\right)} \]

      2. +-commutative 65.1%

         \[\leadsto 1 + \color{blue}{\left(0.20833333333333334 \cdot {x}^{4} + -0.5 \cdot \left(x \cdot x\right)\right)} \]

   6. Applied egg-rr 65.1%

      \[\leadsto 1 + \color{blue}{\left(0.20833333333333334 \cdot {x}^{4} + -0.5 \cdot \left(x \cdot x\right)\right)} \]

   7. Taylor expanded in x around 0 65.1%

      \[\leadsto 1 + \color{blue}{\left(-0.5 \cdot {x}^{2} + 0.20833333333333334 \cdot {x}^{4}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 65.1%

         \[\leadsto 1 + \color{blue}{\left(0.20833333333333334 \cdot {x}^{4} + -0.5 \cdot {x}^{2}\right)} \]

      2. metadata-eval 65.1%

         \[\leadsto 1 + \left(0.20833333333333334 \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}} + -0.5 \cdot {x}^{2}\right) \]

      3. pow-sqr 65.1%

         \[\leadsto 1 + \left(0.20833333333333334 \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} + -0.5 \cdot {x}^{2}\right) \]

      4. unpow2 65.1%

         \[\leadsto 1 + \left(0.20833333333333334 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) + -0.5 \cdot {x}^{2}\right) \]

      5. unpow2 65.1%

         \[\leadsto 1 + \left(0.20833333333333334 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) + -0.5 \cdot {x}^{2}\right) \]

      6. associate-*r* 65.1%

         \[\leadsto 1 + \left(\color{blue}{\left(0.20833333333333334 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)} + -0.5 \cdot {x}^{2}\right) \]

      7. *-commutative 65.1%

         \[\leadsto 1 + \left(\color{blue}{\left(x \cdot x\right) \cdot \left(0.20833333333333334 \cdot \left(x \cdot x\right)\right)} + -0.5 \cdot {x}^{2}\right) \]

      8. unpow2 65.1%

         \[\leadsto 1 + \left(\left(x \cdot x\right) \cdot \left(0.20833333333333334 \cdot \left(x \cdot x\right)\right) + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      9. *-commutative 65.1%

         \[\leadsto 1 + \left(\left(x \cdot x\right) \cdot \left(0.20833333333333334 \cdot \left(x \cdot x\right)\right) + \color{blue}{\left(x \cdot x\right) \cdot -0.5}\right) \]

      10. distribute-lft-out 65.3%

          \[\leadsto 1 + \color{blue}{\left(x \cdot x\right) \cdot \left(0.20833333333333334 \cdot \left(x \cdot x\right) + -0.5\right)} \]

      11. *-commutative 65.3%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot 0.20833333333333334} + -0.5\right) \]

      12. associate-*l* 65.3%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.20833333333333334\right)} + -0.5\right) \]

      13. fma-def 65.3%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right)} \]

   9. Simplified 65.3%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.20833333333333334, -0.5\right)} \]
   10. Step-by-step derivation

       1. fma-udef 65.3%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.20833333333333334\right) + -0.5\right)} \]

   11. Applied egg-rr 65.3%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.20833333333333334\right) + -0.5\right)} \]

   if 340 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 340:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.20833333333333334\right) + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
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 3: 99.2% accurate, 22.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.45) (+ 1.0 (* (* x x) -0.5)) 0.0))

C

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

Fortran

NOTE: x should be positive before calling this function
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

x = Math.abs(x);
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

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

Julia

x = abs(x)
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

x = abs(x)
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

NOTE: x should be positive before calling this function
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 65.2%

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

      1. unpow2 65.2%

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

   4. Simplified 65.2%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 75.4%

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

Alternative 4: 99.0% accurate, 67.3× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 350.0) 1.0 0.0))

C

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

Fortran

NOTE: x should be positive before calling this function
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

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 350.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

x = abs(x)
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

NOTE: x should be positive before calling this function
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 65.2%

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

   if 350 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 75.4%

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

Alternative 5: 51.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.0)

C

x = abs(x);
double code(double x) {
	return 0.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.0;
}

Python

x = abs(x)
def code(x):
	return 0.0

Julia

x = abs(x)
function code(x)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Applied egg-rr 55.7%

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

4. Final simplification 55.7%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))