Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

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: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -1 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 76.6%

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

   1. unpow2 76.6%

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

4. Simplified 76.6%

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

   1. expm1-log1p-u 76.6%

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

   2. expm1-udef 98.7%

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

   3. log1p-udef 98.7%

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

   4. add-exp-log 98.7%

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

   5. +-commutative 98.7%

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

   6. fma-def 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

   \[\leadsto \left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -1 \]

Alternative 3: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {x}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5) (/ 2.0 (+ 2.0 (* x x))) (* 4.0 (pow x -4.0))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 4.0 * pow(x, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = 2.0d0 / (2.0d0 + (x * x))
    else
        tmp = 4.0d0 * (x ** (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 4.0 * Math.pow(x, -4.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = 2.0 / (2.0 + (x * x))
	else:
		tmp = 4.0 * math.pow(x, -4.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * x)));
	else
		tmp = Float64(4.0 * (x ^ -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = 2.0 / (2.0 + (x * x));
	else
		tmp = 4.0 * (x ^ -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.6%

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

      1. unpow2 83.6%

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

   4. Simplified 83.6%

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

   if 3.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 55.7%

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

      1. unpow2 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in x around inf 55.7%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{2}} - 4 \cdot \frac{1}{{x}^{4}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 55.7%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} - 4 \cdot \frac{1}{{x}^{4}} \]

      2. metadata-eval 55.7%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{2}} - 4 \cdot \frac{1}{{x}^{4}} \]

      3. unpow2 55.7%

         \[\leadsto \frac{2}{\color{blue}{x \cdot x}} - 4 \cdot \frac{1}{{x}^{4}} \]

      4. associate-*r/ 55.7%

         \[\leadsto \frac{2}{x \cdot x} - \color{blue}{\frac{4 \cdot 1}{{x}^{4}}} \]

      5. metadata-eval 55.7%

         \[\leadsto \frac{2}{x \cdot x} - \frac{\color{blue}{4}}{{x}^{4}} \]

   7. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{2}{x \cdot x} - \frac{4}{{x}^{4}}} \]

   8. Taylor expanded in x around 0 76.2%

      \[\leadsto \color{blue}{\frac{-4}{{x}^{4}}} \]
   9. Step-by-step derivation

      1. metadata-eval 76.2%

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

      2. pow-sqr 76.2%

         \[\leadsto \frac{-4}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      3. pow-prod-down 76.2%

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

      4. pow2 76.2%

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

      5. associate-/r* 74.5%

         \[\leadsto \color{blue}{\frac{\frac{-4}{x \cdot x}}{x \cdot x}} \]

      6. metadata-eval 74.5%

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

      7. add-sqr-sqrt 74.5%

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

      8. sqrt-unprod 74.5%

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

      9. sqr-neg 74.5%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 74.6%

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

      12. frac-times 74.6%

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

      13. metadata-eval 74.6%

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

      14. frac-2neg 74.6%

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

      15. sqr-neg 74.6%

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

      16. frac-times 74.6%

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

      17. div-inv 74.6%

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

      18. associate-*l* 74.6%

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

   10. Applied egg-rr 74.6%

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

       1. associate-*r* 74.6%

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

       2. metadata-eval 74.6%

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

       3. associate-*r/ 74.6%

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

       4. associate-*r* 74.6%

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

       5. unpow-1 74.6%

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

       6. unpow-1 74.6%

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

       7. pow-sqr 74.6%

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

       8. metadata-eval 74.6%

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

       9. swap-sqr 74.6%

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

       10. metadata-eval 74.6%

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

       11. pow-sqr 74.6%

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

       12. metadata-eval 74.6%

           \[\leadsto 4 \cdot {x}^{\color{blue}{-4}} \]

   12. Simplified 74.6%

       \[\leadsto \color{blue}{4 \cdot {x}^{-4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot {x}^{-4}\\ \end{array} \]

Alternative 4: 82.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{{x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700.0) (/ 2.0 (+ 2.0 (* x x))) (/ -4.0 (pow x 4.0))))

C

double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = -4.0 / pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 700.0d0) then
        tmp = 2.0d0 / (2.0d0 + (x * x))
    else
        tmp = (-4.0d0) / (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 700.0) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = -4.0 / Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 700.0:
		tmp = 2.0 / (2.0 + (x * x))
	else:
		tmp = -4.0 / math.pow(x, 4.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 700.0)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * x)));
	else
		tmp = Float64(-4.0 / (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 700.0)
		tmp = 2.0 / (2.0 + (x * x));
	else
		tmp = -4.0 / (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 700.0], N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 700

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.2%

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

      1. unpow2 83.2%

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

   4. Simplified 83.2%

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

   if 700 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 56.4%

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

      1. unpow2 56.4%

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

   4. Simplified 56.4%

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

   5. Taylor expanded in x around inf 56.4%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{2}} - 4 \cdot \frac{1}{{x}^{4}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 56.4%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{x}^{2}}} - 4 \cdot \frac{1}{{x}^{4}} \]

      2. metadata-eval 56.4%

         \[\leadsto \frac{\color{blue}{2}}{{x}^{2}} - 4 \cdot \frac{1}{{x}^{4}} \]

      3. unpow2 56.4%

         \[\leadsto \frac{2}{\color{blue}{x \cdot x}} - 4 \cdot \frac{1}{{x}^{4}} \]

      4. associate-*r/ 56.4%

         \[\leadsto \frac{2}{x \cdot x} - \color{blue}{\frac{4 \cdot 1}{{x}^{4}}} \]

      5. metadata-eval 56.4%

         \[\leadsto \frac{2}{x \cdot x} - \frac{\color{blue}{4}}{{x}^{4}} \]

   7. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{2}{x \cdot x} - \frac{4}{{x}^{4}}} \]

   8. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{\frac{-4}{{x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{{x}^{4}}\\ \end{array} \]

Alternative 5: 63.3% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.42) 1.0 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		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 <= 1.42d0) 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 <= 1.42) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = 1.0
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = 1.0;
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = 1.0;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], 1.0, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 66.6%

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

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 55.7%

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

      1. unpow2 55.7%

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

   4. Simplified 55.7%

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

   5. Taylor expanded in x around inf 55.7%

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

      1. unpow2 55.7%

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

   7. Simplified 55.7%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \]

Alternative 6: 76.0% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ 2.0 (* x x))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (2.0 + (x * x));
}

Python

def code(x):
	return 2.0 / (2.0 + (x * x))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 76.6%

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

   1. unpow2 76.6%

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

4. Simplified 76.6%

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

5. Final simplification 76.6%

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

Alternative 7: 50.3% 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 50.8%

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

3. Final simplification 50.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))