Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 6

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: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -360.0) 0.0 (if (<= x 360.0) (/ 1.0 (fma 0.5 (* x x) 1.0)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -360.0) {
		tmp = 0.0;
	} else if (x <= 360.0) {
		tmp = 1.0 / fma(0.5, (x * x), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -360.0)
		tmp = 0.0;
	elseif (x <= 360.0)
		tmp = Float64(1.0 / fma(0.5, Float64(x * x), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -360.0], 0.0, If[LessEqual[x, 360.0], N[(1.0 / N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -360 or 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   if -360 < x < 360

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   4. Simplified 98.6%

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

      1. flip-+ 98.6%

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

      2. clear-num 98.6%

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

      3. cancel-sign-sub-inv 98.6%

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

      4. metadata-eval 98.6%

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

      5. metadata-eval 98.6%

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

      6. *-commutative 98.6%

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

      7. *-commutative 98.6%

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

      8. swap-sqr 98.6%

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

      9. pow2 98.6%

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

      10. pow2 98.6%

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

      11. pow-prod-up 98.6%

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

      12. metadata-eval 98.6%

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

      13. metadata-eval 98.6%

          \[\leadsto \frac{1}{\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{1 - {x}^{4} \cdot \color{blue}{0.25}}} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{1 - {x}^{4} \cdot 0.25}}} \]

   7. Taylor expanded in x around 0 98.8%

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

      1. +-commutative 98.8%

         \[\leadsto \frac{1}{\color{blue}{0.5 \cdot {x}^{2} + 1}} \]

      2. unpow2 98.8%

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

      3. fma-udef 98.8%

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

   9. Simplified 98.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 99.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4) 0.0 (if (<= x 1.4) (+ 1.0 (* (* x x) -0.5)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = 0.0;
	} else if (x <= 1.4) {
		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 <= (-1.4d0)) then
        tmp = 0.0d0
    else if (x <= 1.4d0) 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 <= -1.4) {
		tmp = 0.0;
	} else if (x <= 1.4) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.4:
		tmp = 0.0
	elif x <= 1.4:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = 0.0;
	elseif (x <= 1.4)
		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 <= -1.4)
		tmp = 0.0;
	elseif (x <= 1.4)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], 0.0, If[LessEqual[x, 1.4], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 99.2%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. unpow2 99.4%

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

   4. Simplified 99.4%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 59.1% accurate, 40.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -350.0) 0.0 (if (<= x 360.0) 0.5 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -350.0) {
		tmp = 0.0;
	} else 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 <= (-350.0d0)) then
        tmp = 0.0d0
    else 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 <= -350.0) {
		tmp = 0.0;
	} else if (x <= 360.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -350.0:
		tmp = 0.0
	elif x <= 360.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -350.0)
		tmp = 0.0;
	elseif (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 <= -350.0)
		tmp = 0.0;
	elseif (x <= 360.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -350.0], 0.0, If[LessEqual[x, 360.0], 0.5, 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -350 or 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   if -350 < x < 360

   1. Initial program 100.0%

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

   3. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 99.1% accurate, 40.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -350.0) 0.0 (if (<= x 360.0) 1.0 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -350.0) {
		tmp = 0.0;
	} else 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 <= (-350.0d0)) then
        tmp = 0.0d0
    else 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 <= -350.0) {
		tmp = 0.0;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -350.0:
		tmp = 0.0
	elif x <= 360.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -350.0)
		tmp = 0.0;
	elseif (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 <= -350.0)
		tmp = 0.0;
	elseif (x <= 360.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -350.0], 0.0, If[LessEqual[x, 360.0], 1.0, 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -350 or 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

   if -350 < x < 360

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 51.3% 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 50.4%

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

4. Final simplification 50.4%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023193 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))