Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{2}{e^{x} + e^{-x}} \]
4. Add Preprocessing

Alternative 2: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 18000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 2 \cdot {x}^{-2}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 18000.0)
   (/ 2.0 (fma x x 2.0))
   (+ (+ 1.0 (* 2.0 (pow x -2.0))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 18000.0) {
		tmp = 2.0 / fma(x, x, 2.0);
	} else {
		tmp = (1.0 + (2.0 * pow(x, -2.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 18000.0)
		tmp = Float64(2.0 / fma(x, x, 2.0));
	else
		tmp = Float64(Float64(1.0 + Float64(2.0 * (x ^ -2.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 18000.0], N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(2.0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 18000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 87.3%

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

      1. +-commutative 87.3%

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

      2. unpow2 87.3%

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

      3. fma-define 87.3%

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

   5. Simplified 87.3%

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

   if 18000 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. unpow2 55.2%

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

      3. fma-define 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around inf 55.2%

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

      1. expm1-log1p-u 55.2%

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

      2. expm1-undefine 98.2%

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

      3. log1p-undefine 98.2%

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

      4. rem-exp-log 98.2%

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

      5. div-inv 98.2%

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

      6. pow-flip 98.2%

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

      7. metadata-eval 98.2%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 18000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 2 \cdot {x}^{-2}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% 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. Add Preprocessing

3. Taylor expanded in x around 0 80.4%

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

   1. +-commutative 80.4%

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

   2. unpow2 80.4%

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

   3. fma-define 80.4%

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

5. Simplified 80.4%

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

6. Final simplification 80.4%

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

Alternative 4: 63.2% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{--2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.45) 1.0 (/ (- -2.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		tmp = 1.0;
	} else {
		tmp = -(-2.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0
	else:
		tmp = -(-2.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = 1.0;
	else
		tmp = Float64(Float64(-(-2.0)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = 1.0;
	else
		tmp = -(-2.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], 1.0, N[((--2.0) / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.3%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. unpow2 55.2%

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

      3. fma-define 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around inf 55.2%

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

      1. clear-num 55.2%

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

      2. associate-/r/ 55.2%

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

      3. pow-flip 55.2%

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

      4. metadata-eval 55.2%

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

   8. Applied egg-rr 55.2%

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

      1. *-commutative 55.2%

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

      2. sqr-pow 55.2%

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

      3. metadata-eval 55.2%

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

      4. inv-pow 55.2%

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

      5. metadata-eval 55.2%

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

      6. inv-pow 55.2%

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

      7. associate-*r* 55.2%

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

      8. div-inv 55.2%

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

      9. frac-2neg 55.2%

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

      10. metadata-eval 55.2%

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

      11. frac-times 55.2%

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

      12. metadata-eval 55.2%

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

   10. Applied egg-rr 55.2%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{--2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.45) 1.0 (/ (/ 2.0 x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = 1.0
	else:
		tmp = (2.0 / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		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 <= 1.45)
		tmp = 1.0;
	else
		tmp = (2.0 / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], 1.0, N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.3%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. unpow2 55.2%

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

      3. fma-define 55.2%

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

   5. Simplified 55.2%

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

   6. Taylor expanded in x around inf 55.2%

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

      1. add-sqr-sqrt 55.2%

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

      2. sqrt-div 55.2%

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

      3. sqrt-pow1 55.2%

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

      4. metadata-eval 55.2%

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

      5. pow1 55.2%

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

      6. sqrt-div 55.2%

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

      7. sqrt-pow1 55.2%

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

      8. metadata-eval 55.2%

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

      9. pow1 55.2%

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

   8. Applied egg-rr 55.2%

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

      1. associate-*l/ 55.2%

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

      2. associate-*r/ 55.2%

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

      3. rem-square-sqrt 55.2%

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

   10. Simplified 55.2%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in x around 0 58.2%

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

4. Final simplification 58.2%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))