Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

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: 87.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 35000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {x}^{-2} + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 35000.0)
   (/ 2.0 (fma x x 2.0))
   (+ (+ (* 2.0 (pow x -2.0)) 1.0) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 35000.0) {
		tmp = 2.0 / fma(x, x, 2.0);
	} else {
		tmp = ((2.0 * pow(x, -2.0)) + 1.0) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 35000.0)
		tmp = Float64(2.0 / fma(x, x, 2.0));
	else
		tmp = Float64(Float64(Float64(2.0 * (x ^ -2.0)) + 1.0) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 35000.0], N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 35000

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. unpow2 83.1%

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

      3. fma-def 83.1%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   4. Simplified 83.1%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   if 35000 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 45.7%

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

      1. +-commutative 45.7%

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

      2. unpow2 45.7%

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

      3. fma-def 45.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   4. Simplified 45.7%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   5. Taylor expanded in x around inf 45.7%

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

      1. expm1-log1p-u 45.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{{x}^{2}}\right)\right)} \]

      2. expm1-udef 98.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{x}^{2}}\right)} - 1} \]

      3. log1p-udef 98.5%

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

      4. metadata-eval 98.5%

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

      5. add-exp-log 98.5%

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

      6. +-commutative 98.5%

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

      7. div-inv 98.5%

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

      8. pow-flip 98.5%

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

      9. metadata-eval 98.5%

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

      10. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 35000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {x}^{-2} + 1\right) + -1\\ \end{array} \]

Alternative 3: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, x, 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (fma x x 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, x, 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, x, 2.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 73.5%

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

   1. +-commutative 73.5%

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

   2. unpow2 73.5%

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

   3. fma-def 73.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

4. Simplified 73.5%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Final simplification 73.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, x, 2\right)} \]

Alternative 4: 63.2% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{x}{\frac{-1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) 1.0 (/ -2.0 (/ x (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = -2.0 / (x / (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = (-2.0d0) / (x / ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = -2.0 / (x / (-1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = -2.0 / (x / (-1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(-2.0 / Float64(x / Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = -2.0 / (x / (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(-2.0 / N[(x / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 65.6%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 45.7%

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

      1. +-commutative 45.7%

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

      2. unpow2 45.7%

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

      3. fma-def 45.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   4. Simplified 45.7%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   5. Taylor expanded in x around inf 45.7%

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

      1. unpow2 45.7%

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

      2. associate-/r* 45.7%

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

      3. div-inv 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. un-div-inv 45.7%

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

      2. frac-2neg 45.7%

         \[\leadsto \frac{\color{blue}{\frac{-2}{-x}}}{x} \]

      3. metadata-eval 45.7%

         \[\leadsto \frac{\frac{\color{blue}{-2}}{-x}}{x} \]

      4. div-inv 45.7%

         \[\leadsto \frac{\color{blue}{-2 \cdot \frac{1}{-x}}}{x} \]

      5. associate-/l* 45.7%

         \[\leadsto \color{blue}{\frac{-2}{\frac{x}{\frac{1}{-x}}}} \]

      6. neg-mul-1 45.7%

         \[\leadsto \frac{-2}{\frac{x}{\frac{1}{\color{blue}{-1 \cdot x}}}} \]

      7. metadata-eval 45.7%

         \[\leadsto \frac{-2}{\frac{x}{\frac{1}{\color{blue}{\frac{-2}{2}} \cdot x}}} \]

      8. associate-/r* 45.7%

         \[\leadsto \frac{-2}{\frac{x}{\color{blue}{\frac{\frac{1}{\frac{-2}{2}}}{x}}}} \]

      9. metadata-eval 45.7%

         \[\leadsto \frac{-2}{\frac{x}{\frac{\frac{1}{\color{blue}{-1}}}{x}}} \]

      10. metadata-eval 45.7%

          \[\leadsto \frac{-2}{\frac{x}{\frac{\color{blue}{-1}}{x}}} \]

   9. Applied egg-rr 45.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{x}{\frac{-1}{x}}}\\ \end{array} \]

Alternative 5: 63.0% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) 1.0 (/ (/ 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = (2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = (2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 65.6%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 45.7%

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

      1. +-commutative 45.7%

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

      2. unpow2 45.7%

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

      3. fma-def 45.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   4. Simplified 45.7%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   5. Taylor expanded in x around inf 45.7%

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

      1. unpow2 45.7%

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

      2. associate-/r* 45.7%

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

      3. div-inv 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. un-div-inv 45.7%

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

   9. Applied egg-rr 45.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \]

Alternative 6: 50.6% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 49.5%

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

3. Final simplification 49.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))