Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 1.2min

Alternatives: 6

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(2) / ((exp(x)) + (exp((- x))))
END code

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(2) / ((exp(x)) + (exp((- x))))
END code

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\frac{1}{\cosh x} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (/ 1.0 (cosh x)))

C

double code(double x) {
	return 1.0 / cosh(x);
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.0d0 / cosh(x)
end function

Java

public static double code(double x) {
	return 1.0 / Math.cosh(x);
}

Python

def code(x):
	return 1.0 / math.cosh(x)

Julia

function code(x)
	return Float64(1.0 / cosh(x))
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[Cosh[x], $MachinePrecision]), $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	(1) / (((1) / (2)) * ((exp(x)) + ((1) / (exp(x)))))
END code

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \frac{1}{\cosh x} \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \mathsf{fma}\left(x, x, -2\right)\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 2e-6)
  0.0
  (* -0.5 (fma x x -2.0))))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 2e-6) {
		tmp = 0.0;
	} else {
		tmp = -0.5 * fma(x, x, -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 2e-6)
		tmp = 0.0;
	else
		tmp = Float64(-0.5 * fma(x, x, -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-6], 0.0, N[(-0.5 * N[(x * x + -2.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((2) / ((exp(x)) + (exp((- x))))) <= (199999999999999990949622365177251737122787744738161563873291015625e-71)) THEN (0) ELSE ((-5e-1) * ((x * x) + (-2))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 1.9999999999999999e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + \frac{-1}{2} \cdot {x}^{2} \]

   4. Applied rewrites 49.7%

      \[\leadsto 1 + -0.5 \cdot {x}^{2} \]

   6. Applied rewrites 49.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, -2\right), -0.5, 1\right) - 1 \]

   8. Applied rewrites 52.1%

      \[\leadsto 0 \]

   if 1.9999999999999999e-6 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + \frac{-1}{2} \cdot {x}^{2} \]

   4. Applied rewrites 49.7%

      \[\leadsto 1 + -0.5 \cdot {x}^{2} \]

   6. Applied rewrites 49.7%

      \[\leadsto -0.5 \cdot \mathsf{fma}\left(x, x, -2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \leq 1.693928135454124 \cdot 10^{+129}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (+ (exp x) (exp (- x))) 1.693928135454124e+129) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if ((exp(x) + exp(-x)) <= 1.693928135454124e+129) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) + exp(-x)) <= 1.693928135454124d+129) 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)) <= 1.693928135454124e+129) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) + math.exp(-x)) <= 1.693928135454124e+129:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) + exp(Float64(-x))) <= 1.693928135454124e+129)
		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)) <= 1.693928135454124e+129)
		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], 1.693928135454124e+129], 1.0, 0.0]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF (((exp(x)) + (exp((- x)))) <= (1693928135454124127948598477649350140655226641029623547788001815593093350879842447114418741200561560606239272507046029518643920896)) THEN (1) ELSE (0) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 1.6939281354541241e129

   1. Initial program 100.0%

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

   3. Applied rewrites 74.3%

      \[\leadsto \frac{e^{x} \cdot 2}{e^{x + x} - -1} \]

   5. Applied rewrites 50.1%

      \[\leadsto 1 \]

   if 1.6939281354541241e129 < (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 + \frac{-1}{2} \cdot {x}^{2} \]

   4. Applied rewrites 49.7%

      \[\leadsto 1 + -0.5 \cdot {x}^{2} \]

   6. Applied rewrites 49.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, -2\right), -0.5, 1\right) - 1 \]

   8. Applied rewrites 52.1%

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

Alternative 4: 98.6% accurate, 2.2× speedup

Math

\[e^{-\left|x\right|} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (exp (- (fabs x))))

C

double code(double x) {
	return exp(-fabs(x));
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = exp(-abs(x))
end function

Java

public static double code(double x) {
	return Math.exp(-Math.abs(x));
}

Python

def code(x):
	return math.exp(-math.fabs(x))

Julia

function code(x)
	return exp(Float64(-abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = exp(-abs(x));
end

Wolfram

code[x_] := N[Exp[(-N[Abs[x], $MachinePrecision])], $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	exp((- (abs(x))))
END code

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 74.3%

   \[\leadsto \frac{e^{x} \cdot 2}{e^{x + x} - -1} \]

5. Applied rewrites 74.1%

   \[\leadsto e^{-x} \]
6. Add Preprocessing

Alternative 5: 50.1% accurate, 30.4× speedup

Math

\[1 \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    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

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	1
END code

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 74.3%

   \[\leadsto \frac{e^{x} \cdot 2}{e^{x + x} - -1} \]

5. Applied rewrites 50.1%

   \[\leadsto 1 \]
6. Add Preprocessing

Alternative 6: 8.1% accurate, 30.4× speedup

Math

\[0.00033546262790251185 \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  0.00033546262790251185)

C

double code(double x) {
	return 0.00033546262790251185;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.00033546262790251185d0
end function

Java

public static double code(double x) {
	return 0.00033546262790251185;
}

Python

def code(x):
	return 0.00033546262790251185

Julia

function code(x)
	return 0.00033546262790251185
end

MATLAB

function tmp = code(x)
	tmp = 0.00033546262790251185;
end

Wolfram

code[x_] := 0.00033546262790251185

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	3354626279025118532235716362066568763111717998981475830078125e-64
END code

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 74.3%

   \[\leadsto \frac{e^{x} \cdot 2}{e^{x + x} - -1} \]

5. Applied rewrites 73.3%

   \[\leadsto e^{x} \]

7. Applied rewrites 8.1%

   \[\leadsto e^{-8} \]

8. Evaluated real constant 8.1%

   \[\leadsto 0.00033546262790251185 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2026074 +o generate:egglog
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))