Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

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: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{2 + \left(x \cdot x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 2e+16)
   (/ 2.0 (+ 2.0 (* (* x x) (+ 1.0 (* (* x x) 0.08333333333333333)))))
   0.0))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 2e+16) {
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) + exp(-x)) <= 2d+16) then
        tmp = 2.0d0 / (2.0d0 + ((x * x) * (1.0d0 + ((x * x) * 0.08333333333333333d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) + Math.exp(-x)) <= 2e+16) {
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) + math.exp(-x)) <= 2e+16:
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 2e+16)
		tmp = Float64(2.0 / Float64(2.0 + Float64(Float64(x * x) * Float64(1.0 + Float64(Float64(x * x) * 0.08333333333333333)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) + exp(-x)) <= 2e+16)
		tmp = 2.0 / (2.0 + ((x * x) * (1.0 + ((x * x) * 0.08333333333333333))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e+16], N[(2.0 / N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right)} \]

   5. Simplified 98.4%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)}} \]
   6. Step-by-step derivation

      1. unpow2 98.4%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   7. Applied egg-rr 98.4%

      \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]
   8. Step-by-step derivation

      1. unpow2 98.4%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   9. Applied egg-rr 98.4%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)} \]

   if 2e16 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 2e+16) (/ 2.0 (+ 2.0 (* x x))) 0.0))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 2e+16) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) + exp(-x)) <= 2d+16) then
        tmp = 2.0d0 / (2.0d0 + (x * x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) + Math.exp(-x)) <= 2e+16) {
		tmp = 2.0 / (2.0 + (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) + math.exp(-x)) <= 2e+16:
		tmp = 2.0 / (2.0 + (x * x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 2e+16)
		tmp = Float64(2.0 / Float64(2.0 + Float64(x * x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) + exp(-x)) <= 2e+16)
		tmp = 2.0 / (2.0 + (x * x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2e+16], N[(2.0 / N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.4%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right)} \]

   5. Simplified 98.4%

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)}} \]
   6. Step-by-step derivation

      1. unpow2 98.4%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   7. Applied egg-rr 98.4%

      \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]
   8. Step-by-step derivation

      1. unpow2 98.4%

         \[\leadsto \frac{2}{2 + {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right)} \]

   9. Applied egg-rr 98.4%

      \[\leadsto \frac{2}{2 + \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot 0.08333333333333333\right)} \]

   10. Taylor expanded in x around 0 98.1%

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

   if 2e16 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{2 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 1.8e+154) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 1.8e+154) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) + exp(-x)) <= 1.8d+154) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) + Math.exp(-x)) <= 1.8e+154) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) + math.exp(-x)) <= 1.8e+154:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 1.8e+154)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) + exp(-x)) <= 1.8e+154)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 1.8e+154], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 1.8e154

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.6%

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

   if 1.8e154 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 2.8 \cdot 10^{+154}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (exp x) (exp (- x))) 2.8e+154) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 2.8e+154) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) + exp(-x)) <= 2.8d+154) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) + Math.exp(-x)) <= 2.8e+154) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) + math.exp(-x)) <= 2.8e+154:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 2.8e+154)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) + exp(-x)) <= 2.8e+154)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.8e+154], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.7999999999999999e154

   1. Initial program 100.0%

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

   4. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{0.5} \]

   if 2.7999999999999999e154 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + \left(1 - x\right)} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + (1.0d0 - x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + (1.0 - x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + (1.0 - x))

Julia

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

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + (1.0 - x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 73.8%

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

   1. mul-1-neg 73.8%

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

   2. unsub-neg 73.8%

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

5. Simplified 73.8%

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

Alternative 7: 51.1% 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%

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

4. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))