NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 98.8%

Time: 18.6s

Alternatives: 9

Speedup: 1.1×

• 37.7% 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: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + eps\_m\right)\\ \frac{e^{x \cdot \left(-1 + eps\_m\right)} + {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{t\_0 \cdot t\_0}\right)}\right)}^{\left(\sqrt[3]{x \cdot eps\_m}\right)}}{2} \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 eps_m))))
   (/
    (+
     (exp (* x (+ -1.0 eps_m)))
     (pow (pow (exp -1.0) (cbrt (* t_0 t_0))) (cbrt (* x eps_m))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	return (exp((x * (-1.0 + eps_m))) + pow(pow(exp(-1.0), cbrt((t_0 * t_0))), cbrt((x * eps_m)))) / 2.0;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	return (Math.exp((x * (-1.0 + eps_m))) + Math.pow(Math.pow(Math.exp(-1.0), Math.cbrt((t_0 * t_0))), Math.cbrt((x * eps_m)))) / 2.0;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + eps_m))
	return Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + ((exp(-1.0) ^ cbrt(Float64(t_0 * t_0))) ^ cbrt(Float64(x * eps_m)))) / 2.0)
end

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in eps around inf 97.8%

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

   1. exp-prod 97.8%

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

   2. add-cube-cbrt 97.8%

      \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left(e^{-1}\right)}^{\color{blue}{\left(\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right) \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}}{2} \]

   3. pow-unpow 97.8%

      \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot \color{blue}{{\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)} \cdot \sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}}{2} \]

   4. cbrt-unprod 97.8%

      \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\color{blue}{\left(\sqrt[3]{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}{2} \]

   5. pow2 97.8%

      \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\color{blue}{{\left(x \cdot \left(1 + \varepsilon\right)\right)}^{2}}}\right)}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}{2} \]

7. Applied egg-rr 97.8%

   \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot \color{blue}{{\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{{\left(x \cdot \left(1 + \varepsilon\right)\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}}{2} \]
8. Step-by-step derivation

   1. unpow2 97.8%

      \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}{2} \]

9. Applied egg-rr 97.8%

   \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)}\right)}^{\left(\sqrt[3]{x \cdot \left(1 + \varepsilon\right)}\right)}}{2} \]

10. Taylor expanded in eps around inf 89.7%

    \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)}^{\left(\sqrt[3]{\color{blue}{\varepsilon \cdot x}}\right)}}{2} \]
11. Step-by-step derivation

    1. *-commutative 89.7%

       \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)}^{\left(\sqrt[3]{\color{blue}{x \cdot \varepsilon}}\right)}}{2} \]

12. Simplified 89.7%

    \[\leadsto \frac{e^{-1 \cdot \left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)}^{\left(\sqrt[3]{\color{blue}{x \cdot \varepsilon}}\right)}}{2} \]

13. Final simplification 89.7%

    \[\leadsto \frac{e^{x \cdot \left(-1 + \varepsilon\right)} + {\left({\left(e^{-1}\right)}^{\left(\sqrt[3]{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)}^{\left(\sqrt[3]{x \cdot \varepsilon}\right)}}{2} \]
14. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (pow(exp(-1.0), (x * (1.0 - eps_m))) + exp((x * (-1.0 - eps_m)))) / 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((-1.0d0)) ** (x * (1.0d0 - eps_m))) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in eps around inf 97.8%

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

   1. exp-prod 97.8%

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

7. Applied egg-rr 97.8%

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

8. Final simplification 97.8%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (-1.0 + eps_m))) + exp((x * (-1.0 - eps_m)))) / 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 * ((-1.0d0) - eps_m)))) / 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 * (-1.0 - eps_m)))) / 2.0;
}

Python

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

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (-1.0 + eps_m))) + exp((x * (-1.0 - eps_m)))) / 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[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in eps around inf 97.8%

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

6. Final simplification 97.8%

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

Alternative 4: 91.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (-1.0 + eps_m))) + exp(-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 * ((-1.0d0) + eps_m))) + exp(-x)) / 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)) / 2.0;
}

Python

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

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 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[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in eps around inf 97.8%

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

6. Taylor expanded in eps around 0 82.1%

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

   1. neg-mul-1 82.1%

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

8. Simplified 82.1%

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

9. Final simplification 82.1%

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

Alternative 5: 63.4% accurate, 15.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (((1.0 - x) - (eps_m * (x * -2.0))) + (x + 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 = (((1.0d0 - x) - (eps_m * (x * (-2.0d0)))) + (x + 1.0d0)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in x around 0 40.6%

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

6. Taylor expanded in x around 0 26.5%

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

   1. add-sqr-sqrt 13.4%

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

   2. sqrt-unprod 28.0%

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

   3. sqr-neg 28.0%

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

   4. sqrt-unprod 12.6%

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

   5. add-sqr-sqrt 25.1%

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

   6. distribute-lft-neg-in 25.1%

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

8. Applied egg-rr 25.1%

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

9. Taylor expanded in eps around -inf 60.9%

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

    1. sub-neg 60.9%

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

    2. associate-+r+ 60.9%

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

    3. mul-1-neg 60.9%

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

    4. unsub-neg 60.9%

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

    5. mul-1-neg 60.9%

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

    6. unsub-neg 60.9%

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

    7. *-lft-identity 60.9%

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

    8. distribute-rgt-out-- 60.9%

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

    9. metadata-eval 60.9%

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

    10. mul-1-neg 60.9%

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

    11. remove-double-neg 60.9%

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

    12. +-commutative 60.9%

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

11. Simplified 60.9%

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

12. Final simplification 60.9%

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

Alternative 6: 57.3% accurate, 25.2× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return ((1.0 - x) + (x + 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 = ((1.0d0 - x) + (x + 1.0d0)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in x around 0 40.6%

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

6. Taylor expanded in x around 0 26.5%

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

7. Taylor expanded in eps around 0 56.2%

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

   1. sub-neg 56.2%

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

   2. +-commutative 56.2%

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

   3. mul-1-neg 56.2%

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

   4. remove-double-neg 56.2%

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

   5. neg-mul-1 56.2%

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

   6. unsub-neg 56.2%

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

9. Simplified 56.2%

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

10. Final simplification 56.2%

    \[\leadsto \frac{\left(1 - x\right) + \left(x + 1\right)}{2} \]
11. Add Preprocessing

Alternative 7: 10.1% accurate, 32.4× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (eps_m * (x + 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 = (eps_m * (x + x)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in x around 0 40.6%

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

6. Taylor expanded in x around 0 26.5%

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

   1. add-sqr-sqrt 13.4%

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

   2. sqrt-unprod 28.0%

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

   3. sqr-neg 28.0%

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

   4. sqrt-unprod 12.6%

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

   5. add-sqr-sqrt 25.1%

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

   6. distribute-lft-neg-in 25.1%

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

8. Applied egg-rr 25.1%

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

9. Taylor expanded in eps around inf 8.9%

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

    1. cancel-sign-sub-inv 8.9%

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

    2. metadata-eval 8.9%

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

    3. *-lft-identity 8.9%

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

11. Simplified 8.9%

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

12. Final simplification 8.9%

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

Alternative 8: 10.1% accurate, 45.4× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (x * eps_m) / 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 = (x * eps_m) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in x around 0 40.6%

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

6. Taylor expanded in x around inf 8.8%

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

   1. mul-1-neg 8.8%

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

   2. *-commutative 8.8%

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

   3. associate-*l* 8.8%

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

   4. distribute-rgt-neg-in 8.8%

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

   5. distribute-neg-in 8.8%

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

   6. metadata-eval 8.8%

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

   7. distribute-neg-frac 8.8%

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

   8. metadata-eval 8.8%

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

8. Simplified 8.8%

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

9. Taylor expanded in eps around inf 8.9%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
10. Step-by-step derivation

    1. *-commutative 8.9%

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

11. Simplified 8.9%

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

12. Final simplification 8.9%

    \[\leadsto \frac{x \cdot \varepsilon}{2} \]
13. Add Preprocessing

Alternative 9: 44.7% 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 76.5%

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

   1. fma-neg 76.5%

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

   2. /-rgt-identity 76.5%

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

   3. fma-neg 76.5%

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

   4. /-rgt-identity 76.5%

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

   5. distribute-rgt-neg-in 76.5%

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

   6. sub-neg 76.5%

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

   7. metadata-eval 76.5%

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

   8. distribute-rgt-neg-in 76.5%

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

3. Simplified 76.5%

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

5. Taylor expanded in x around 0 41.5%

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

6. Final simplification 41.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(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))