expq2 (section 3.11)

Percentage Accurate: 37.1% → 100.0%

Time: 4.6s

Alternatives: 8

Speedup: 68.3×

Specification

\[710 > x\]

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{expm1}\left(x\right)} \cdot e^{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 (expm1 x)) (exp x)))

C

double code(double x) {
	return (1.0 / expm1(x)) * exp(x);
}

Java

public static double code(double x) {
	return (1.0 / Math.expm1(x)) * Math.exp(x);
}

Python

def code(x):
	return (1.0 / math.expm1(x)) * math.exp(x)

Julia

function code(x)
	return Float64(Float64(1.0 / expm1(x)) * exp(x))
end

Wolfram

code[x_] := N[(N[(1.0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

double code(double x) {
	return exp(x) / expm1(x);
}

Java

public static double code(double x) {
	return Math.exp(x) / Math.expm1(x);
}

Python

def code(x):
	return math.exp(x) / math.expm1(x)

Julia

function code(x)
	return Float64(exp(x) / expm1(x))
end

Wolfram

code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(-0.001388888888888889 \cdot {x}^{3} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (/ (exp x) x)
   (+
    0.5
    (+
     (* -0.001388888888888889 (pow x 3.0))
     (+ (* x 0.08333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = exp(x) / x;
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.8d0)) then
        tmp = exp(x) / x
    else
        tmp = 0.5d0 + (((-0.001388888888888889d0) * (x ** 3.0d0)) + ((x * 0.08333333333333333d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = Math.exp(x) / x;
	} else {
		tmp = 0.5 + ((-0.001388888888888889 * Math.pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.8:
		tmp = math.exp(x) / x
	else:
		tmp = 0.5 + ((-0.001388888888888889 * math.pow(x, 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(exp(x) / x);
	else
		tmp = Float64(0.5 + Float64(Float64(-0.001388888888888889 * (x ^ 3.0)) + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.8)
		tmp = exp(x) / x;
	else
		tmp = 0.5 + ((-0.001388888888888889 * (x ^ 3.0)) + ((x * 0.08333333333333333) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.8], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(0.5 + N[(N[(-0.001388888888888889 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0022:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0022)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0022) {
		tmp = 1.0 / (1.0 - exp(-x));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0022d0)) then
        tmp = 1.0d0 / (1.0d0 - exp(-x))
    else
        tmp = 0.5d0 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0022) {
		tmp = 1.0 / (1.0 - Math.exp(-x));
	} else {
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0022:
		tmp = 1.0 / (1.0 - math.exp(-x))
	else:
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0022)
		tmp = Float64(1.0 / Float64(1.0 - exp(Float64(-x))));
	else
		tmp = Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0022)
		tmp = 1.0 / (1.0 - exp(-x));
	else
		tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0022], N[(1.0 / N[(1.0 - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) x))

C

double code(double x) {
	return exp(x) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) / x
end function

Java

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

Python

def code(x):
	return math.exp(x) / x

Julia

function code(x)
	return Float64(exp(x) / x)
end

MATLAB

function tmp = code(x)
	tmp = exp(x) / x;
end

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 67.3% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x))))

C

double code(double x) {
	return 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + ((x * 0.08333333333333333d0) + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
}

Python

def code(x):
	return 0.5 + ((x * 0.08333333333333333) + (1.0 / x))

Julia

function code(x)
	return Float64(0.5 + Float64(Float64(x * 0.08333333333333333) + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + ((x * 0.08333333333333333) + (1.0 / x));
end

Wolfram

code[x_] := N[(0.5 + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 67.2% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 0.5 (/ 1.0 x)))

C

double code(double x) {
	return 0.5 + (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + (1.0d0 / x)
end function

Java

public static double code(double x) {
	return 0.5 + (1.0 / x);
}

Python

def code(x):
	return 0.5 + (1.0 / x)

Julia

function code(x)
	return Float64(0.5 + Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + (1.0 / x);
end

Wolfram

code[x_] := N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 67.3% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 x))

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

code[x_] := N[((-1.0) / N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :herbie-target
  (/ (- 1.0) (expm1 (- x)))

  (/ (exp x) (- (exp x) 1.0)))