NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.9%

Time: 12.1s

Alternatives: 14

Speedup: 2.1×

• 38.3% 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: 73.5% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := e^{-x}\\ t_2 := t\_1 + x \cdot t\_1\\ t_3 := e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right)\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(eps\_m + -1\right)} + t\_3 \leq 2:\\ \;\;\;\;\frac{t\_2 + t\_2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{eps\_m \cdot x} + t\_3}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m)))
        (t_1 (exp (- x)))
        (t_2 (+ t_1 (* x t_1)))
        (t_3 (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m)))))
   (if (<= (+ (* t_0 (exp (* x (+ eps_m -1.0)))) t_3) 2.0)
     (/ (+ t_2 t_2) 2.0)
     (/ (+ (* t_0 (exp (* eps_m x))) t_3) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = exp(-x);
	double t_2 = t_1 + (x * t_1);
	double t_3 = exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m));
	double tmp;
	if (((t_0 * exp((x * (eps_m + -1.0)))) + t_3) <= 2.0) {
		tmp = (t_2 + t_2) / 2.0;
	} else {
		tmp = ((t_0 * exp((eps_m * x))) + t_3) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 + (1.0d0 / eps_m)
    t_1 = exp(-x)
    t_2 = t_1 + (x * t_1)
    t_3 = exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m))
    if (((t_0 * exp((x * (eps_m + (-1.0d0))))) + t_3) <= 2.0d0) then
        tmp = (t_2 + t_2) / 2.0d0
    else
        tmp = ((t_0 * exp((eps_m * x))) + t_3) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = Math.exp(-x);
	double t_2 = t_1 + (x * t_1);
	double t_3 = Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m));
	double tmp;
	if (((t_0 * Math.exp((x * (eps_m + -1.0)))) + t_3) <= 2.0) {
		tmp = (t_2 + t_2) / 2.0;
	} else {
		tmp = ((t_0 * Math.exp((eps_m * x))) + t_3) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (1.0 / eps_m)
	t_1 = math.exp(-x)
	t_2 = t_1 + (x * t_1)
	t_3 = math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m))
	tmp = 0
	if ((t_0 * math.exp((x * (eps_m + -1.0)))) + t_3) <= 2.0:
		tmp = (t_2 + t_2) / 2.0
	else:
		tmp = ((t_0 * math.exp((eps_m * x))) + t_3) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_1 + Float64(x * t_1))
	t_3 = Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps_m + -1.0)))) + t_3) <= 2.0)
		tmp = Float64(Float64(t_2 + t_2) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(eps_m * x))) + t_3) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (1.0 / eps_m);
	t_1 = exp(-x);
	t_2 = t_1 + (x * t_1);
	t_3 = exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m));
	tmp = 0.0;
	if (((t_0 * exp((x * (eps_m + -1.0)))) + t_3) <= 2.0)
		tmp = (t_2 + t_2) / 2.0;
	else
		tmp = ((t_0 * exp((eps_m * x))) + t_3) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], 2.0], N[(N[(t$95$2 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2

   1. Initial program 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in eps around 0 100.0%

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

   if 2 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in eps around inf 97.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;\frac{\left(e^{-x} + x \cdot e^{-x}\right) + \left(e^{-x} + x \cdot e^{-x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := 2 \cdot e^{-x}\\ t_2 := e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(1 + \frac{-1}{eps\_m}\right)\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(eps\_m + -1\right)} + t\_2 \leq 2:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(t\_1 + \frac{t\_1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{eps\_m \cdot x} + t\_2}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m)))
        (t_1 (* 2.0 (exp (- x))))
        (t_2 (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m)))))
   (if (<= (+ (* t_0 (exp (* x (+ eps_m -1.0)))) t_2) 2.0)
     (* x (* 0.5 (+ t_1 (/ t_1 x))))
     (/ (+ (* t_0 (exp (* eps_m x))) t_2) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = 2.0 * exp(-x);
	double t_2 = exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m));
	double tmp;
	if (((t_0 * exp((x * (eps_m + -1.0)))) + t_2) <= 2.0) {
		tmp = x * (0.5 * (t_1 + (t_1 / x)));
	} else {
		tmp = ((t_0 * exp((eps_m * x))) + t_2) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (1.0d0 / eps_m)
    t_1 = 2.0d0 * exp(-x)
    t_2 = exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m))
    if (((t_0 * exp((x * (eps_m + (-1.0d0))))) + t_2) <= 2.0d0) then
        tmp = x * (0.5d0 * (t_1 + (t_1 / x)))
    else
        tmp = ((t_0 * exp((eps_m * x))) + t_2) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = 2.0 * Math.exp(-x);
	double t_2 = Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m));
	double tmp;
	if (((t_0 * Math.exp((x * (eps_m + -1.0)))) + t_2) <= 2.0) {
		tmp = x * (0.5 * (t_1 + (t_1 / x)));
	} else {
		tmp = ((t_0 * Math.exp((eps_m * x))) + t_2) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (1.0 / eps_m)
	t_1 = 2.0 * math.exp(-x)
	t_2 = math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m))
	tmp = 0
	if ((t_0 * math.exp((x * (eps_m + -1.0)))) + t_2) <= 2.0:
		tmp = x * (0.5 * (t_1 + (t_1 / x)))
	else:
		tmp = ((t_0 * math.exp((eps_m * x))) + t_2) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(2.0 * exp(Float64(-x)))
	t_2 = Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps_m + -1.0)))) + t_2) <= 2.0)
		tmp = Float64(x * Float64(0.5 * Float64(t_1 + Float64(t_1 / x))));
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(eps_m * x))) + t_2) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (1.0 / eps_m);
	t_1 = 2.0 * exp(-x);
	t_2 = exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m));
	tmp = 0.0;
	if (((t_0 * exp((x * (eps_m + -1.0)))) + t_2) <= 2.0)
		tmp = x * (0.5 * (t_1 + (t_1 / x)));
	else
		tmp = ((t_0 * exp((eps_m * x))) + t_2) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], 2.0], N[(x * N[(0.5 * N[(t$95$1 + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2

   1. Initial program 50.6%

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

   3. Simplified 50.6%

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

   5. Taylor expanded in eps around 0 100.0%

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

   6. Taylor expanded in x around inf 99.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right) + 0.5 \cdot \frac{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}{x}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}{x} + 0.5 \cdot \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)} \]

      2. distribute-lft-out 99.9%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\frac{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right)} \]

      3. cancel-sign-sub-inv 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{\color{blue}{e^{-1 \cdot x} + \left(--1\right) \cdot e^{-1 \cdot x}}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      4. neg-mul-1 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{e^{\color{blue}{-x}} + \left(--1\right) \cdot e^{-1 \cdot x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      5. metadata-eval 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{e^{-x} + \color{blue}{1} \cdot e^{-1 \cdot x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      6. neg-mul-1 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{e^{-x} + 1 \cdot e^{\color{blue}{-x}}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      7. distribute-rgt1-in 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{\color{blue}{\left(1 + 1\right) \cdot e^{-x}}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      8. metadata-eval 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{\color{blue}{2} \cdot e^{-x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      9. cancel-sign-sub-inv 99.9%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{2 \cdot e^{-x}}{x} + \color{blue}{\left(e^{-1 \cdot x} + \left(--1\right) \cdot e^{-1 \cdot x}\right)}\right)\right) \]

      10. neg-mul-1 99.9%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{2 \cdot e^{-x}}{x} + \left(e^{\color{blue}{-x}} + \left(--1\right) \cdot e^{-1 \cdot x}\right)\right)\right) \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(\frac{2 \cdot e^{-x}}{x} + 2 \cdot e^{-x}\right)\right)} \]

   if 2 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in eps around inf 97.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(2 \cdot e^{-x} + \frac{2 \cdot e^{-x}}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 2 \cdot e^{-x}\\ \mathbf{if}\;eps\_m \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(t\_0 + \frac{t\_0}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(eps\_m \cdot x\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* 2.0 (exp (- x)))))
   (if (<= eps_m 9.5e-12)
     (* x (* 0.5 (+ t_0 (/ t_0 x))))
     (/ (* 2.0 (cosh (* eps_m x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 2.0 * exp(-x);
	double tmp;
	if (eps_m <= 9.5e-12) {
		tmp = x * (0.5 * (t_0 + (t_0 / x)));
	} else {
		tmp = (2.0 * cosh((eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * exp(-x)
    if (eps_m <= 9.5d-12) then
        tmp = x * (0.5d0 * (t_0 + (t_0 / x)))
    else
        tmp = (2.0d0 * cosh((eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 2.0 * Math.exp(-x);
	double tmp;
	if (eps_m <= 9.5e-12) {
		tmp = x * (0.5 * (t_0 + (t_0 / x)));
	} else {
		tmp = (2.0 * Math.cosh((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 2.0 * math.exp(-x)
	tmp = 0
	if eps_m <= 9.5e-12:
		tmp = x * (0.5 * (t_0 + (t_0 / x)))
	else:
		tmp = (2.0 * math.cosh((eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(2.0 * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 9.5e-12)
		tmp = Float64(x * Float64(0.5 * Float64(t_0 + Float64(t_0 / x))));
	else
		tmp = Float64(Float64(2.0 * cosh(Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 2.0 * exp(-x);
	tmp = 0.0;
	if (eps_m <= 9.5e-12)
		tmp = x * (0.5 * (t_0 + (t_0 / x)));
	else
		tmp = (2.0 * cosh((eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 9.5e-12], N[(x * N[(0.5 * N[(t$95$0 + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.4999999999999995e-12

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in eps around 0 66.1%

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

   6. Taylor expanded in x around inf 66.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right) + 0.5 \cdot \frac{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}{x}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 66.0%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}{x} + 0.5 \cdot \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)} \]

      2. distribute-lft-out 66.0%

         \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\frac{e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right)} \]

      3. cancel-sign-sub-inv 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{\color{blue}{e^{-1 \cdot x} + \left(--1\right) \cdot e^{-1 \cdot x}}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      4. neg-mul-1 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{e^{\color{blue}{-x}} + \left(--1\right) \cdot e^{-1 \cdot x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      5. metadata-eval 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{e^{-x} + \color{blue}{1} \cdot e^{-1 \cdot x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      6. neg-mul-1 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{e^{-x} + 1 \cdot e^{\color{blue}{-x}}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      7. distribute-rgt1-in 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{\color{blue}{\left(1 + 1\right) \cdot e^{-x}}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      8. metadata-eval 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{\color{blue}{2} \cdot e^{-x}}{x} + \left(e^{-1 \cdot x} - -1 \cdot e^{-1 \cdot x}\right)\right)\right) \]

      9. cancel-sign-sub-inv 66.0%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{2 \cdot e^{-x}}{x} + \color{blue}{\left(e^{-1 \cdot x} + \left(--1\right) \cdot e^{-1 \cdot x}\right)}\right)\right) \]

      10. neg-mul-1 66.0%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\frac{2 \cdot e^{-x}}{x} + \left(e^{\color{blue}{-x}} + \left(--1\right) \cdot e^{-1 \cdot x}\right)\right)\right) \]

   8. Simplified 66.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(\frac{2 \cdot e^{-x}}{x} + 2 \cdot e^{-x}\right)\right)} \]

   if 9.4999999999999995e-12 < eps

   1. Initial program 99.8%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in eps around inf 99.8%

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

       1. *-commutative 99.8%

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

   11. Simplified 99.8%

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

       1. +-commutative 99.8%

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

       2. distribute-lft-neg-out 99.8%

          \[\leadsto \frac{e^{x \cdot \varepsilon} + e^{\color{blue}{-\varepsilon \cdot x}}}{2} \]

       3. *-commutative 99.8%

          \[\leadsto \frac{e^{x \cdot \varepsilon} + e^{-\color{blue}{x \cdot \varepsilon}}}{2} \]

       4. cosh-undef 99.9%

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

   13. Applied egg-rr 99.9%

       \[\leadsto \frac{\color{blue}{2 \cdot \cosh \left(x \cdot \varepsilon\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(2 \cdot e^{-x} + \frac{2 \cdot e^{-x}}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \cosh \left(\varepsilon \cdot x\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (- -1.0 eps_m))) (exp (* x (+ eps_m -1.0)))) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

3. Simplified 61.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 96.9%

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

6. Final simplification 96.9%

   \[\leadsto \frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \left(\varepsilon + -1\right)}}{2} \]
7. Add Preprocessing

Alternative 5: 91.8% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

3. Simplified 61.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 96.9%

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

6. Taylor expanded in eps around inf 85.9%

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

   1. associate-*r* 85.9%

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

   2. neg-mul-1 85.9%

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

8. Simplified 85.9%

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

9. Final simplification 85.9%

   \[\leadsto \frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2} \]
10. Add Preprocessing

Alternative 6: 70.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 470.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 5e+199) 0.0 (+ 1.0 (* (* x x) (* x 0.3333333333333333))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 470.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 5e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 470.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 5d+199) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + ((x * x) * (x * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 470.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 5e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 470.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 5e+199:
		tmp = 0.0
	else:
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 470.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 5e+199)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(x * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 470.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 5e+199)
		tmp = 0.0;
	else
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 470.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+199], 0.0, N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 470

   1. Initial program 59.1%

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

   3. Simplified 44.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 95.6%

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

   6. Taylor expanded in eps around inf 95.6%

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

      1. associate-*r* 95.6%

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

      2. neg-mul-1 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in eps around 0 72.7%

      \[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot x}}}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 72.7%

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

   11. Simplified 72.7%

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

   if 470 < x < 4.9999999999999998e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 53.2%

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

      1. associate-*r* 53.2%

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

      2. neg-mul-1 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in eps around inf 47.3%

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

       1. *-commutative 47.3%

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

   11. Simplified 47.3%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 54.7%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 4.9999999999999998e199 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.0%

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

   6. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   7. Step-by-step derivation

      1. unpow2 63.5%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   8. Applied egg-rr 63.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   9. Taylor expanded in x around inf 63.5%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 63.5%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]

   11. Simplified 63.5%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 \cdot \cosh \left(eps\_m \cdot x\right)}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (/ (* 2.0 (cosh (* eps_m x))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 * cosh((eps_m * x))) / 2.0;
}

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (2.0 * Math.cosh((eps_m * x))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 * math.cosh((eps_m * x))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 * cosh(Float64(eps_m * x))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 * cosh((eps_m * x))) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(2.0 * N[Cosh[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.4%

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

3. Simplified 61.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 96.9%

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

6. Taylor expanded in eps around inf 85.9%

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

   1. associate-*r* 85.9%

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

   2. neg-mul-1 85.9%

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

8. Simplified 85.9%

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

9. Taylor expanded in eps around inf 83.3%

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

    1. *-commutative 83.3%

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

11. Simplified 83.3%

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

    1. +-commutative 83.3%

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

    2. distribute-lft-neg-out 83.3%

       \[\leadsto \frac{e^{x \cdot \varepsilon} + e^{\color{blue}{-\varepsilon \cdot x}}}{2} \]

    3. *-commutative 83.3%

       \[\leadsto \frac{e^{x \cdot \varepsilon} + e^{-\color{blue}{x \cdot \varepsilon}}}{2} \]

    4. cosh-undef 83.3%

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

13. Applied egg-rr 83.3%

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

14. Final simplification 83.3%

    \[\leadsto \frac{2 \cdot \cosh \left(\varepsilon \cdot x\right)}{2} \]
15. Add Preprocessing

Alternative 8: 57.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;1 - \left(x \cdot x\right) \cdot \left(0.5 - x \cdot \left(0.3333333333333333 + x \cdot -0.125\right)\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.8)
   (- 1.0 (* (* x x) (- 0.5 (* x (+ 0.3333333333333333 (* x -0.125))))))
   (if (<= x 5e+199) 0.0 (+ 1.0 (* (* x x) (* x 0.3333333333333333))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 - ((x * x) * (0.5 - (x * (0.3333333333333333 + (x * -0.125)))));
	} else if (x <= 5e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = 1.0d0 - ((x * x) * (0.5d0 - (x * (0.3333333333333333d0 + (x * (-0.125d0))))))
    else if (x <= 5d+199) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + ((x * x) * (x * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 - ((x * x) * (0.5 - (x * (0.3333333333333333 + (x * -0.125)))));
	} else if (x <= 5e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.8:
		tmp = 1.0 - ((x * x) * (0.5 - (x * (0.3333333333333333 + (x * -0.125)))))
	elif x <= 5e+199:
		tmp = 0.0
	else:
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(1.0 - Float64(Float64(x * x) * Float64(0.5 - Float64(x * Float64(0.3333333333333333 + Float64(x * -0.125))))));
	elseif (x <= 5e+199)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(x * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = 1.0 - ((x * x) * (0.5 - (x * (0.3333333333333333 + (x * -0.125)))));
	elseif (x <= 5e+199)
		tmp = 0.0;
	else
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.8], N[(1.0 - N[(N[(x * x), $MachinePrecision] * N[(0.5 - N[(x * N[(0.3333333333333333 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+199], 0.0, N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.80000000000000004

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around 0 59.1%

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

   6. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(x \cdot \left(0.3333333333333333 + -0.125 \cdot x\right) - 0.5\right)} \]
   7. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   8. Applied egg-rr 58.7%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(0.3333333333333333 + -0.125 \cdot x\right) - 0.5\right) \]

   if 1.80000000000000004 < x < 4.9999999999999998e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 53.2%

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

      1. associate-*r* 53.2%

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

      2. neg-mul-1 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in eps around inf 47.3%

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

       1. *-commutative 47.3%

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

   11. Simplified 47.3%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 54.7%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 4.9999999999999998e199 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.0%

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

   6. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   7. Step-by-step derivation

      1. unpow2 63.5%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   8. Applied egg-rr 63.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   9. Taylor expanded in x around inf 63.5%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 63.5%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]

   11. Simplified 63.5%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;1 - \left(x \cdot x\right) \cdot \left(0.5 - x \cdot \left(0.3333333333333333 + x \cdot -0.125\right)\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 490.0)
   (+ 1.0 (* (* x x) (- (* x 0.3333333333333333) 0.5)))
   (if (<= x 3.7e+199) 0.0 (+ 1.0 (* (* x x) (* x 0.3333333333333333))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 3.7e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 490.0d0) then
        tmp = 1.0d0 + ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0))
    else if (x <= 3.7d+199) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + ((x * x) * (x * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 3.7e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 490.0:
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5))
	elif x <= 3.7e+199:
		tmp = 0.0
	else:
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 490.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)));
	elseif (x <= 3.7e+199)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(x * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 490.0)
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	elseif (x <= 3.7e+199)
		tmp = 0.0;
	else
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 490.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e+199], 0.0, N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 490

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around 0 59.1%

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

   6. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   7. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   8. Applied egg-rr 58.7%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   if 490 < x < 3.70000000000000021e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 53.2%

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

      1. associate-*r* 53.2%

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

      2. neg-mul-1 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in eps around inf 47.3%

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

       1. *-commutative 47.3%

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

   11. Simplified 47.3%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 54.7%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 3.70000000000000021e199 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.0%

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

   6. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   7. Step-by-step derivation

      1. unpow2 63.5%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   8. Applied egg-rr 63.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   9. Taylor expanded in x around inf 63.5%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 63.5%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]

   11. Simplified 63.5%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.42)
   (/ (- 2.0 (* x x)) 2.0)
   (if (<= x 4e+199) 0.0 (+ 1.0 (* (* x x) (* x 0.3333333333333333))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 4e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 4d+199) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + ((x * x) * (x * 0.3333333333333333d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 4e+199) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 4e+199:
		tmp = 0.0
	else:
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 4e+199)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(x * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 4e+199)
		tmp = 0.0;
	else
		tmp = 1.0 + ((x * x) * (x * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+199], 0.0, N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.4199999999999999

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around 0 59.1%

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

   6. Taylor expanded in x around 0 58.5%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot {x}^{2}}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 58.5%

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

      2. unsub-neg 58.5%

         \[\leadsto \frac{\color{blue}{2 - {x}^{2}}}{2} \]

   8. Simplified 58.5%

      \[\leadsto \frac{\color{blue}{2 - {x}^{2}}}{2} \]
   9. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 58.5%

       \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]

   if 1.4199999999999999 < x < 4.00000000000000039e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 53.2%

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

      1. associate-*r* 53.2%

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

      2. neg-mul-1 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in eps around inf 47.3%

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

       1. *-commutative 47.3%

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

   11. Simplified 47.3%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 54.7%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 4.00000000000000039e199 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.0%

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

   6. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   7. Step-by-step derivation

      1. unpow2 63.5%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   8. Applied egg-rr 63.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   9. Taylor expanded in x around inf 63.5%

      \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 63.5%

          \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]

   11. Simplified 63.5%

       \[\leadsto 1 + \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.3333333333333333\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.42)
   (/ (- 2.0 (* x x)) 2.0)
   (if (<= x 4e+199) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 4e+199) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 4d+199) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 4e+199) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 4e+199:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 4e+199)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 4e+199)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+199], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.4199999999999999

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around 0 59.1%

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

   6. Taylor expanded in x around 0 58.5%

      \[\leadsto \frac{\color{blue}{2 + -1 \cdot {x}^{2}}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 58.5%

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

      2. unsub-neg 58.5%

         \[\leadsto \frac{\color{blue}{2 - {x}^{2}}}{2} \]

   8. Simplified 58.5%

      \[\leadsto \frac{\color{blue}{2 - {x}^{2}}}{2} \]
   9. Step-by-step derivation

      1. unpow2 58.7%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 58.5%

       \[\leadsto \frac{2 - \color{blue}{x \cdot x}}{2} \]

   if 1.4199999999999999 < x < 4.00000000000000039e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 53.2%

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

      1. associate-*r* 53.2%

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

      2. neg-mul-1 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in eps around inf 47.3%

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

       1. *-commutative 47.3%

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

   11. Simplified 47.3%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 54.7%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 4.00000000000000039e199 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.0%

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

   7. Taylor expanded in eps around inf 34.5%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 480.0) 1.0 (if (<= x 3.7e+199) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 3.7e+199) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 480.0d0) then
        tmp = 1.0d0
    else if (x <= 3.7d+199) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 3.7e+199) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 480.0:
		tmp = 1.0
	elif x <= 3.7e+199:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 3.7e+199)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 3.7e+199)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 3.7e+199], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 480

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around inf 56.2%

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

   6. Taylor expanded in x around 0 18.2%

      \[\leadsto \frac{\color{blue}{2 - \frac{1}{\varepsilon}}}{2} \]

   7. Taylor expanded in eps around inf 58.3%

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

   if 480 < x < 3.70000000000000021e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 53.2%

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

      1. associate-*r* 53.2%

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

      2. neg-mul-1 53.2%

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

   8. Simplified 53.2%

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

   9. Taylor expanded in eps around inf 47.3%

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

       1. *-commutative 47.3%

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

   11. Simplified 47.3%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 54.7%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 54.7%

       \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 3.70000000000000021e199 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.0%

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

   7. Taylor expanded in eps around inf 34.5%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+199}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 490.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 490.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 490.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 490.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 490

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around inf 56.2%

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

   6. Taylor expanded in x around 0 18.2%

      \[\leadsto \frac{\color{blue}{2 - \frac{1}{\varepsilon}}}{2} \]

   7. Taylor expanded in eps around inf 58.3%

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

   if 490 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 63.3%

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

      1. associate-*r* 63.3%

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

      2. neg-mul-1 63.3%

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

   8. Simplified 63.3%

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

   9. Taylor expanded in eps around inf 53.0%

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

       1. *-commutative 53.0%

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

   11. Simplified 53.0%

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

       1. flip-+ 0.1%

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

       2. div-sub 0.1%

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

   13. Applied egg-rr 0.0%

       \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}} - \frac{{\left({\left(e^{\varepsilon}\right)}^{x}\right)}^{2}}{{\left(e^{\varepsilon}\right)}^{x} - {\left(e^{\varepsilon}\right)}^{x}}}}{2} \]
   14. Step-by-step derivation

       1. +-inverses 48.9%

          \[\leadsto \frac{\color{blue}{0}}{2} \]

   15. Simplified 48.9%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.6% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 71.4%

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

3. Simplified 71.4%

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

5. Taylor expanded in eps around inf 69.4%

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

6. Taylor expanded in x around 0 13.6%

   \[\leadsto \frac{\color{blue}{2 - \frac{1}{\varepsilon}}}{2} \]

7. Taylor expanded in eps around inf 41.7%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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))