NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 98.7%

Time: 14.1s

Alternatives: 14

Speedup: 1.1×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (pow(exp(-1.0), (x * (1.0 - eps))) + exp((x * (-1.0 - eps)))) / 2.0;
}

Fortran

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

Java

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

Python

def code(x, eps):
	return (math.pow(math.exp(-1.0), (x * (1.0 - eps))) + math.exp((x * (-1.0 - eps)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64((exp(-1.0) ^ Float64(x * Float64(1.0 - eps))) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((exp(-1.0) ^ (x * (1.0 - eps))) + exp((x * (-1.0 - eps)))) / 2.0;
end

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 2: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 3: 76.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-232)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps))) (exp (- x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.2d-232)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-232:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps))) + math.exp(-x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-232)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-232)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-232], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 4: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(-1 + \varepsilon\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-232)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (/ (+ 1.0 (pow E (* x (+ -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (1.0 + pow(((double) M_E), (x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (1.0 + Math.pow(Math.E, (x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-232:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (1.0 + math.pow(math.e, (x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-232)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-232)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-232], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Power[E, N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 5: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-232)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.2d-232)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-232:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-232)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-232)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-232], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 6: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.2e-232)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.2d-232)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.2e-232) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.2e-232:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.2e-232)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.2e-232)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.2e-232], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 7: 56.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -16500000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 30000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -16500000.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 30000000000.0) 1.0 (/ (/ (expm1 x) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -16500000.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 30000000000.0) {
		tmp = 1.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -16500000.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 30000000000.0) {
		tmp = 1.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -16500000.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 30000000000.0:
		tmp = 1.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -16500000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 30000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -16500000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 30000000000.0], 1.0, N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 8: 63.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 30000000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 30000000000.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (/ (expm1 x) eps) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 30000000000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 30000000000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 30000000000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 30000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 30000000000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 9: 54.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq 9500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - t_0\right) + 0.5 \cdot \left(t_0 \cdot t_0\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 eps))))
   (if (<= eps 9500.0)
     1.0
     (/ (+ (+ (- 1.0 t_0) (* 0.5 (* t_0 t_0))) (+ 1.0 (/ -1.0 eps))) 2.0))))

C

double code(double x, double eps) {
	double t_0 = x * (1.0 - eps);
	double tmp;
	if (eps <= 9500.0) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - t_0) + (0.5 * (t_0 * t_0))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 - eps)
    if (eps <= 9500.0d0) then
        tmp = 1.0d0
    else
        tmp = (((1.0d0 - t_0) + (0.5d0 * (t_0 * t_0))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (1.0 - eps);
	double tmp;
	if (eps <= 9500.0) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - t_0) + (0.5 * (t_0 * t_0))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (1.0 - eps)
	tmp = 0
	if eps <= 9500.0:
		tmp = 1.0
	else:
		tmp = (((1.0 - t_0) + (0.5 * (t_0 * t_0))) + (1.0 + (-1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(1.0 - eps))
	tmp = 0.0
	if (eps <= 9500.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_0) + Float64(0.5 * Float64(t_0 * t_0))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (1.0 - eps);
	tmp = 0.0;
	if (eps <= 9500.0)
		tmp = 1.0;
	else
		tmp = (((1.0 - t_0) + (0.5 * (t_0 * t_0))) + (1.0 + (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 9500.0], 1.0, N[(N[(N[(N[(1.0 - t$95$0), $MachinePrecision] + N[(0.5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 10: 56.0% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(1 - \varepsilon\right)\\ \mathbf{if}\;\varepsilon \leq 9000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - t_0\right) + 0.5 \cdot \left(x \cdot \left(\left(1 - \varepsilon\right) \cdot t_0\right)\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 eps))))
   (if (<= eps 9000.0)
     1.0
     (/
      (+
       (+ (- 1.0 t_0) (* 0.5 (* x (* (- 1.0 eps) t_0))))
       (+ 1.0 (/ -1.0 eps)))
      2.0))))

C

double code(double x, double eps) {
	double t_0 = x * (1.0 - eps);
	double tmp;
	if (eps <= 9000.0) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - t_0) + (0.5 * (x * ((1.0 - eps) * t_0)))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 - eps)
    if (eps <= 9000.0d0) then
        tmp = 1.0d0
    else
        tmp = (((1.0d0 - t_0) + (0.5d0 * (x * ((1.0d0 - eps) * t_0)))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (1.0 - eps);
	double tmp;
	if (eps <= 9000.0) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - t_0) + (0.5 * (x * ((1.0 - eps) * t_0)))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (1.0 - eps)
	tmp = 0
	if eps <= 9000.0:
		tmp = 1.0
	else:
		tmp = (((1.0 - t_0) + (0.5 * (x * ((1.0 - eps) * t_0)))) + (1.0 + (-1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(1.0 - eps))
	tmp = 0.0
	if (eps <= 9000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - t_0) + Float64(0.5 * Float64(x * Float64(Float64(1.0 - eps) * t_0)))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (1.0 - eps);
	tmp = 0.0;
	if (eps <= 9000.0)
		tmp = 1.0;
	else
		tmp = (((1.0 - t_0) + (0.5 * (x * ((1.0 - eps) * t_0)))) + (1.0 + (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 9000.0], 1.0, N[(N[(N[(N[(1.0 - t$95$0), $MachinePrecision] + N[(0.5 * N[(x * N[(N[(1.0 - eps), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 11: 53.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-1}{\varepsilon}\\ \mathbf{if}\;x \leq -0.0012:\\ \;\;\;\;\frac{2 - x \cdot \left(\left(1 + \varepsilon\right) \cdot t_0\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -1.0 eps))))
   (if (<= x -0.0012)
     (/ (- 2.0 (* x (* (+ 1.0 eps) t_0))) 2.0)
     (if (<= x 1.6e-11)
       1.0
       (if (<= x 1.45e+83)
         (/ (+ (- 1.0 (/ -1.0 eps)) t_0) 2.0)
         (/ (+ 2.0 (* x eps)) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (-1.0 / eps);
	double tmp;
	if (x <= -0.0012) {
		tmp = (2.0 - (x * ((1.0 + eps) * t_0))) / 2.0;
	} else if (x <= 1.6e-11) {
		tmp = 1.0;
	} else if (x <= 1.45e+83) {
		tmp = ((1.0 - (-1.0 / eps)) + t_0) / 2.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-1.0d0) / eps)
    if (x <= (-0.0012d0)) then
        tmp = (2.0d0 - (x * ((1.0d0 + eps) * t_0))) / 2.0d0
    else if (x <= 1.6d-11) then
        tmp = 1.0d0
    else if (x <= 1.45d+83) then
        tmp = ((1.0d0 - ((-1.0d0) / eps)) + t_0) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (-1.0 / eps);
	double tmp;
	if (x <= -0.0012) {
		tmp = (2.0 - (x * ((1.0 + eps) * t_0))) / 2.0;
	} else if (x <= 1.6e-11) {
		tmp = 1.0;
	} else if (x <= 1.45e+83) {
		tmp = ((1.0 - (-1.0 / eps)) + t_0) / 2.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + (-1.0 / eps)
	tmp = 0
	if x <= -0.0012:
		tmp = (2.0 - (x * ((1.0 + eps) * t_0))) / 2.0
	elif x <= 1.6e-11:
		tmp = 1.0
	elif x <= 1.45e+83:
		tmp = ((1.0 - (-1.0 / eps)) + t_0) / 2.0
	else:
		tmp = (2.0 + (x * eps)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(-1.0 / eps))
	tmp = 0.0
	if (x <= -0.0012)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(1.0 + eps) * t_0))) / 2.0);
	elseif (x <= 1.6e-11)
		tmp = 1.0;
	elseif (x <= 1.45e+83)
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps)) + t_0) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 + (-1.0 / eps);
	tmp = 0.0;
	if (x <= -0.0012)
		tmp = (2.0 - (x * ((1.0 + eps) * t_0))) / 2.0;
	elseif (x <= 1.6e-11)
		tmp = 1.0;
	elseif (x <= 1.45e+83)
		tmp = ((1.0 - (-1.0 / eps)) + t_0) / 2.0;
	else
		tmp = (2.0 + (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0012], N[(N[(2.0 - N[(x * N[(N[(1.0 + eps), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e-11], 1.0, If[LessEqual[x, 1.45e+83], N[(N[(N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

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

Alternative 12: 50.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+85}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.6e-11)
   1.0
   (if (<= x 1e+85)
     (/ (+ (- 1.0 (/ -1.0 eps)) (+ 1.0 (/ -1.0 eps))) 2.0)
     (/ (+ 2.0 (* x eps)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.6e-11) {
		tmp = 1.0;
	} else if (x <= 1e+85) {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.6d-11) then
        tmp = 1.0d0
    else if (x <= 1d+85) then
        tmp = ((1.0d0 - ((-1.0d0) / eps)) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.6e-11) {
		tmp = 1.0;
	} else if (x <= 1e+85) {
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.6e-11:
		tmp = 1.0
	elif x <= 1e+85:
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0
	else:
		tmp = (2.0 + (x * eps)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.6e-11)
		tmp = 1.0;
	elseif (x <= 1e+85)
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps)) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.6e-11)
		tmp = 1.0;
	elseif (x <= 1e+85)
		tmp = ((1.0 - (-1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	else
		tmp = (2.0 + (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.6e-11], 1.0, If[LessEqual[x, 1e+85], N[(N[(N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 13: 50.6% accurate, 32.4× speedup

Math

\[\begin{array}{l} \\ \frac{2 + x \cdot \varepsilon}{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (+ 2.0 (* x eps)) 2.0))

C

double code(double x, double eps) {
	return (2.0 + (x * eps)) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 + (x * eps)) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 + (x * eps)) / 2.0;
}

Python

def code(x, eps):
	return (2.0 + (x * eps)) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(2.0 + Float64(x * eps)) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 + (x * eps)) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

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

Alternative 14: 44.1% accurate, 227.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

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

Reproduce

herbie shell --seed 2023347 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))