expq2 (section 3.11)

Percentage Accurate: 37.6% → 100.0%

Time: 6.8s

Alternatives: 9

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: 37.6% 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

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

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 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0023:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x \cdot 0.08333333333333333 + 0.5\right)\right) + \left(\frac{1}{x} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0023)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+ (+ 1.0 (+ (* x 0.08333333333333333) 0.5)) (+ (/ 1.0 x) -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0023) {
		tmp = 1.0 / (1.0 - exp(-x));
	} else {
		tmp = (1.0 + ((x * 0.08333333333333333) + 0.5)) + ((1.0 / x) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0023d0)) then
        tmp = 1.0d0 / (1.0d0 - exp(-x))
    else
        tmp = (1.0d0 + ((x * 0.08333333333333333d0) + 0.5d0)) + ((1.0d0 / x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0023) {
		tmp = 1.0 / (1.0 - Math.exp(-x));
	} else {
		tmp = (1.0 + ((x * 0.08333333333333333) + 0.5)) + ((1.0 / x) + -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0023:
		tmp = 1.0 / (1.0 - math.exp(-x))
	else:
		tmp = (1.0 + ((x * 0.08333333333333333) + 0.5)) + ((1.0 / x) + -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0023)
		tmp = Float64(1.0 / Float64(1.0 - exp(Float64(-x))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(x * 0.08333333333333333) + 0.5)) + Float64(Float64(1.0 / x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0023)
		tmp = 1.0 / (1.0 - exp(-x));
	else
		tmp = (1.0 + ((x * 0.08333333333333333) + 0.5)) + ((1.0 / x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0023], N[(1.0 / N[(1.0 - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(x * 0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 3: 98.1% 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 4: 67.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 5: 67.0% 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 6: 66.9% 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 7: 66.9% 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

Alternative 8: 3.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

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

Alternative 9: 3.3% accurate, 205.0× 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

&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 2024006 
(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)))