Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 24000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{2}{{x}^{2}}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 24000.0)
   (/ 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 <= 24000.0) {
		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 <= 24000.0)
		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, 24000.0], 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 < 24000

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. fma-define 85.7%

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

   5. Simplified 85.7%

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

   if 24000 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. fma-define 51.5%

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

   5. Simplified 51.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 51.5%

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

      2. expm1-undefine 100.0%

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

      3. log1p-undefine 100.0%

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

      4. rem-exp-log 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 24000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{2}{{x}^{2}}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = 1.0
	else:
		tmp = 2.0 / math.pow(x, 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = 1.0;
	else
		tmp = Float64(2.0 / (x ^ 2.0));
	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 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], 1.0, N[(2.0 / N[Power[x, 2.0], $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. Add Preprocessing

   3. Taylor expanded in x around 0 68.6%

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

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.5%

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

      1. +-commutative 50.5%

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

      2. unpow2 50.5%

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

      3. fma-define 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in x around inf 50.5%

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

Alternative 4: 98.5% 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. Add Preprocessing

3. Taylor expanded in x around 0 76.9%

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

   1. +-commutative 76.9%

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

   2. unpow2 76.9%

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

   3. fma-define 76.9%

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

5. Simplified 76.9%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 76.9%

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

   2. expm1-undefine 97.9%

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

   3. log1p-undefine 97.9%

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

   4. rem-exp-log 97.9%

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

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

Alternative 5: 75.9% 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 76.9%

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

   1. +-commutative 76.9%

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

   2. unpow2 76.9%

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

   3. fma-define 76.9%

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

5. Simplified 76.9%

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

Alternative 6: 63.3% accurate, 20.6× speedup

Math

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

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(Float64(2.0 / 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[(N[(2.0 / x), $MachinePrecision] / x), $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. Add Preprocessing

   3. Taylor expanded in x around 0 68.6%

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

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.5%

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

      1. +-commutative 50.5%

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

      2. unpow2 50.5%

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

      3. fma-define 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in x around inf 50.5%

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

      1. clear-num 50.5%

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

      2. associate-/r/ 50.5%

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

      3. pow-flip 50.0%

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

      4. metadata-eval 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. *-commutative 50.0%

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

      2. sqr-pow 50.0%

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

      3. sqr-pow 50.0%

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

      4. metadata-eval 50.0%

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

      5. pow-flip 50.5%

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

      6. div-inv 50.5%

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

      7. unpow2 50.5%

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

      8. associate-/r* 50.0%

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

   10. Applied egg-rr 50.0%

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

Alternative 7: 51.2% 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 51.3%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))