NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.8%

Time: 11.5s

Alternatives: 8

Speedup: 0.5×

• 38.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -4.159413150046609e+207
  2. eps = -1.3755621304025788e-271

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: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $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 55.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 100.0%

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

   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 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-inverses N/A

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

      7. div-add N/A

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

      8. +-commutative N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      4.0)
   (/ (+ 1.0 x) (exp x))
   (/ (- (/ (+ 1.0 eps) eps) (- (exp (- (fma x eps x))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (1.0 + x) / exp(x);
	} else {
		tmp = (((1.0 + eps) / eps) - -exp(-fma(x, eps, x))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + eps) / eps) - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $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))))) < 4

   1. Initial program 55.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

   if 4 < (-.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 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 56.7

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

   10. Applied rewrites 56.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \varepsilon}{\varepsilon}\\ \mathbf{if}\;x \leq -2500000000:\\ \;\;\;\;\frac{t\_0 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 eps) eps)))
   (if (<= x -2500000000.0)
     (/ (- t_0 (- (fma x eps x) 1.0)) 2.0)
     (if (<= x 1.8)
       (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
       (* (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + eps) / eps;
	double tmp;
	if (x <= -2500000000.0) {
		tmp = (t_0 - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (x <= 1.8) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (0.5 * x) * fma((eps - 1.0), t_0, pow(eps, -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + eps) / eps)
	tmp = 0.0
	if (x <= -2500000000.0)
		tmp = Float64(Float64(t_0 - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (x <= 1.8)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(0.5 * x) * fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision]}, If[LessEqual[x, -2500000000.0], N[(N[(t$95$0 - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in x around 0

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

       1. *-inverses N/A

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

       2. div-add N/A

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

       3. +-commutative N/A

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

       4. lower-/.f64 N/A

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

       5. lower-+.f64 36.3

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

   11. Applied rewrites 36.3%

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

   if -2.5e9 < x < 1.80000000000000004

   1. Initial program 48.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(-x, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 79.5%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon} - 1, \left(\varepsilon - 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right), 1\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.7%

      \[\leadsto \left(0.5 \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 63.0%

       \[\leadsto \left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \frac{1}{\varepsilon}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500000000:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, {\varepsilon}^{-1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(2 + x\right) + x\right) \cdot \mathsf{fma}\left(0.25 \cdot x - 0.5, x, 0.5\right)\\ \mathbf{if}\;\varepsilon \leq -110000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ (+ 2.0 x) x) (fma (- (* 0.25 x) 0.5) x 0.5))))
   (if (<= eps -110000000.0)
     t_0
     (if (<= eps 1.0)
       (/ (+ 1.0 x) (exp x))
       (if (<= eps 4.6e+157)
         t_0
         (*
          (* 0.5 x)
          (fma
           (- eps 1.0)
           (/ (+ 1.0 eps) eps)
           (/ (- 1.0 (* eps eps)) eps))))))))

C

double code(double x, double eps) {
	double t_0 = ((2.0 + x) + x) * fma(((0.25 * x) - 0.5), x, 0.5);
	double tmp;
	if (eps <= -110000000.0) {
		tmp = t_0;
	} else if (eps <= 1.0) {
		tmp = (1.0 + x) / exp(x);
	} else if (eps <= 4.6e+157) {
		tmp = t_0;
	} else {
		tmp = (0.5 * x) * fma((eps - 1.0), ((1.0 + eps) / eps), ((1.0 - (eps * eps)) / eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(2.0 + x) + x) * fma(Float64(Float64(0.25 * x) - 0.5), x, 0.5))
	tmp = 0.0
	if (eps <= -110000000.0)
		tmp = t_0;
	elseif (eps <= 1.0)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	elseif (eps <= 4.6e+157)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * x) * fma(Float64(eps - 1.0), Float64(Float64(1.0 + eps) / eps), Float64(Float64(1.0 - Float64(eps * eps)) / eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision] * x + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -110000000.0], t$95$0, If[LessEqual[eps, 1.0], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.6e+157], t$95$0, N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.1e8 or 1 < eps < 4.60000000000000008e157

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 31.6%

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

   7. Applied rewrites 31.6%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 31.6%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left(\left(2 + x\right) + x\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} \cdot x - \frac{1}{2}\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 52.9%

       \[\leadsto \left(\left(2 + x\right) + x\right) \cdot \mathsf{fma}\left(0.25 \cdot x - 0.5, \color{blue}{x}, 0.5\right) \]

   if -1.1e8 < eps < 1

   1. Initial program 44.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.2%

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

   if 4.60000000000000008e157 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon} - 1, \left(\varepsilon - 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right), 1\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.6%

      \[\leadsto \left(0.5 \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 66.7%

       \[\leadsto \left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(2 + x\right) + x\right) \cdot \mathsf{fma}\left(0.25 \cdot x - 0.5, x, 0.5\right)\\ \mathbf{if}\;\varepsilon \leq -110000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\varepsilon \leq 520000:\\ \;\;\;\;\frac{\left(x + 2\right) + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)} \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{+157}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ (+ 2.0 x) x) (fma (- (* 0.25 x) 0.5) x 0.5))))
   (if (<= eps -110000000.0)
     t_0
     (if (<= eps 520000.0)
       (*
        (/
         (+ (+ x 2.0) x)
         (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
        0.5)
       (if (<= eps 4.6e+157)
         t_0
         (*
          (* 0.5 x)
          (fma
           (- eps 1.0)
           (/ (+ 1.0 eps) eps)
           (/ (- 1.0 (* eps eps)) eps))))))))

C

double code(double x, double eps) {
	double t_0 = ((2.0 + x) + x) * fma(((0.25 * x) - 0.5), x, 0.5);
	double tmp;
	if (eps <= -110000000.0) {
		tmp = t_0;
	} else if (eps <= 520000.0) {
		tmp = (((x + 2.0) + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0)) * 0.5;
	} else if (eps <= 4.6e+157) {
		tmp = t_0;
	} else {
		tmp = (0.5 * x) * fma((eps - 1.0), ((1.0 + eps) / eps), ((1.0 - (eps * eps)) / eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(2.0 + x) + x) * fma(Float64(Float64(0.25 * x) - 0.5), x, 0.5))
	tmp = 0.0
	if (eps <= -110000000.0)
		tmp = t_0;
	elseif (eps <= 520000.0)
		tmp = Float64(Float64(Float64(Float64(x + 2.0) + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0)) * 0.5);
	elseif (eps <= 4.6e+157)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 * x) * fma(Float64(eps - 1.0), Float64(Float64(1.0 + eps) / eps), Float64(Float64(1.0 - Float64(eps * eps)) / eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(0.25 * x), $MachinePrecision] - 0.5), $MachinePrecision] * x + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -110000000.0], t$95$0, If[LessEqual[eps, 520000.0], N[(N[(N[(N[(x + 2.0), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 4.6e+157], t$95$0, N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.1e8 or 5.2e5 < eps < 4.60000000000000008e157

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 29.9%

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

   7. Applied rewrites 29.9%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 29.9%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left(\left(2 + x\right) + x\right) \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\frac{1}{4} \cdot x - \frac{1}{2}\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 52.0%

       \[\leadsto \left(\left(2 + x\right) + x\right) \cdot \mathsf{fma}\left(0.25 \cdot x - 0.5, \color{blue}{x}, 0.5\right) \]

   if -1.1e8 < eps < 5.2e5

   1. Initial program 46.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \cdot \frac{1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.2%

       \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)} \cdot 0.5 \]

   if 4.60000000000000008e157 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon} - 1, \left(\varepsilon - 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right), 1\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 1.6%

      \[\leadsto \left(0.5 \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 66.7%

       \[\leadsto \left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -110000000:\\ \;\;\;\;\left(\left(2 + x\right) + x\right) \cdot \mathsf{fma}\left(0.25 \cdot x - 0.5, x, 0.5\right)\\ \mathbf{elif}\;\varepsilon \leq 520000:\\ \;\;\;\;\frac{\left(x + 2\right) + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)} \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{+157}:\\ \;\;\;\;\left(\left(2 + x\right) + x\right) \cdot \mathsf{fma}\left(0.25 \cdot x - 0.5, x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2500000000:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2500000000.0)
   (/ (- (/ (+ 1.0 eps) eps) (- (fma x eps x) 1.0)) 2.0)
   (if (<= x 1.8)
     (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
     (* (* 0.5 x) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2500000000.0) {
		tmp = (((1.0 + eps) / eps) - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (x <= 1.8) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (0.5 * x) * 0.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2500000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + eps) / eps) - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (x <= 1.8)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(0.5 * x) * 0.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2500000000.0], N[(N[(N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in x around 0

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

       1. *-inverses N/A

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

       2. div-add N/A

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

       3. +-commutative N/A

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

       4. lower-/.f64 N/A

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

       5. lower-+.f64 36.3

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

   11. Applied rewrites 36.3%

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

   if -2.5e9 < x < 1.80000000000000004

   1. Initial program 48.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(-x, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 79.5%

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

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon} - 1, \left(\varepsilon - 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right), 1\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.7%

      \[\leadsto \left(0.5 \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 55.9%

       \[\leadsto \left(0.5 \cdot x\right) \cdot 0 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.8% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 440:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 440.0) 1.0 (* (* 0.5 x) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 440.0) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * x) * 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 440.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 * x) * 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 440.0) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * x) * 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 440.0:
		tmp = 1.0
	else:
		tmp = (0.5 * x) * 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 440.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 * x) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 440.0)
		tmp = 1.0;
	else
		tmp = (0.5 * x) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 440.0], 1.0, N[(N[(0.5 * x), $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 440

   1. Initial program 59.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.2%

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

   if 440 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(1 + \varepsilon, \frac{1}{\varepsilon} - 1, \left(\varepsilon - 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right), 1\right)} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.7%

      \[\leadsto \left(0.5 \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, \frac{1 + \varepsilon}{\varepsilon}, \left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 55.9%

       \[\leadsto \left(0.5 \cdot x\right) \cdot 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 43.7% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 42.1%

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

Reproduce

herbie shell --seed 2024326 
(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))