NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.9%

Time: 14.6s

Alternatives: 15

Speedup: 9.7×

• 38.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.0559996326651763e+148
  2. eps = -2.2502935795700683e-97

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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + t\_0 \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 + e^{x \cdot \varepsilon}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps)))))
   (if (<=
        (+
         (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
         (* t_0 (+ 1.0 (/ -1.0 eps))))
        0.0)
     (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
     (* 0.5 (+ t_0 (exp (* x eps)))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (t_0 + exp((x * eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) - eps)))
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (t_0 * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = 0.5d0 * (exp(-x) * (x + (x + 2.0d0)))
    else
        tmp = 0.5d0 * (t_0 + exp((x * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * (-1.0 - eps)));
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (Math.exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (t_0 + Math.exp((x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * (-1.0 - eps)))
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = 0.5 * (math.exp(-x) * (x + (x + 2.0)))
	else:
		tmp = 0.5 * (t_0 + math.exp((x * eps)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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))))) < 0.0

   1. Initial program 31.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-out-- N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   if 0.0 < (-.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

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto 0.5 \cdot \left(e^{\color{blue}{x \cdot \varepsilon}} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) - x\right)\right) \cdot \left(0.5 \cdot x\right), \frac{1}{\varepsilon}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      2.0)
   (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
   (fma (* (* eps (- (* eps (* x eps)) x)) (* 0.5 x)) (/ 1.0 eps) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.5%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-out-- N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 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

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon} + 1 \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon} + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + \color{blue}{x \cdot \varepsilon}\right)\right)}{\varepsilon} + 1 \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \color{blue}{\left(0 + x \cdot \varepsilon\right)}\right)}{\varepsilon} + 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\varepsilon \cdot \left(0 + x \cdot \varepsilon\right)}\right)}{\varepsilon} + 1 \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \color{blue}{\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      8. +-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{\color{blue}{\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      9. div-inv N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right) \cdot \frac{1}{\varepsilon}\right)} + 1 \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right)\right) \cdot \frac{1}{\varepsilon}} + 1 \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right), \frac{1}{\varepsilon}, 1\right)} \]

   10. Applied rewrites 92.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) - x\right)\right) \cdot \left(x \cdot 0.5\right), \frac{1}{\varepsilon}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

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

Alternative 3: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) - x\right)\right) \cdot \left(0.5 \cdot x\right), \frac{1}{\varepsilon}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      2.0)
   (exp (- x))
   (fma (* (* eps (- (* eps (* x eps)) x)) (* 0.5 x)) (/ 1.0 eps) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.5%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{-1 \cdot x}} \cdot 1 \]

      7. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-1 \cdot x}} \cdot 1 \]

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 97.9

         \[\leadsto e^{\color{blue}{-x}} \cdot 1 \]

   8. Applied rewrites 97.9%

      \[\leadsto \color{blue}{e^{-x} \cdot 1} \]

   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 x around 0

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon} + 1 \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon} + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + \color{blue}{x \cdot \varepsilon}\right)\right)}{\varepsilon} + 1 \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \color{blue}{\left(0 + x \cdot \varepsilon\right)}\right)}{\varepsilon} + 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\varepsilon \cdot \left(0 + x \cdot \varepsilon\right)}\right)}{\varepsilon} + 1 \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \color{blue}{\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      8. +-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{\color{blue}{\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      9. div-inv N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right) \cdot \frac{1}{\varepsilon}\right)} + 1 \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right)\right) \cdot \frac{1}{\varepsilon}} + 1 \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right), \frac{1}{\varepsilon}, 1\right)} \]

   10. Applied rewrites 92.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) - x\right)\right) \cdot \left(x \cdot 0.5\right), \frac{1}{\varepsilon}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

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

Alternative 4: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)
   (* 0.5 (* x (* x (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 51.5%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-out-- N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{3} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 77.6

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

   8. Applied rewrites 77.6%

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

   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 x around 0

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 91.6%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 87.4

           \[\leadsto 0.5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Applied rewrites 87.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{1}{e^{\mathsf{fma}\left(x, -\varepsilon, x\right)}} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* 0.5 (+ (/ 1.0 (exp (fma x (- eps) x))) (exp (* x (- -1.0 eps))))))

C

double code(double x, double eps) {
	return 0.5 * ((1.0 / exp(fma(x, -eps, x))) + exp((x * (-1.0 - eps))));
}

Julia

function code(x, eps)
	return Float64(0.5 * Float64(Float64(1.0 / exp(fma(x, Float64(-eps), x))) + exp(Float64(x * Float64(-1.0 - eps)))))
end

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 98.8%

   \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{1}{e^{\mathsf{fma}\left(x, -\varepsilon, x\right)}}} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]
8. Add Preprocessing

Alternative 6: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(e^{\mathsf{fma}\left(x, \varepsilon, -x\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* 0.5 (+ (exp (fma x eps (- x))) (exp (* x (- -1.0 eps))))))

C

double code(double x, double eps) {
	return 0.5 * (exp(fma(x, eps, -x)) + exp((x * (-1.0 - eps))));
}

Julia

function code(x, eps)
	return Float64(0.5 * Float64(exp(fma(x, eps, Float64(-x))) + exp(Float64(x * Float64(-1.0 - eps)))))
end

Wolfram

code[x_, eps_] := N[(0.5 * N[(N[Exp[N[(x * eps + (-x)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 98.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{\mathsf{fma}\left(x, \varepsilon, -x\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \]
6. Add Preprocessing

Alternative 7: 87.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) - x\right)\right) \cdot \left(0.5 \cdot x\right), \frac{1}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 9.5e-5)
   (fma (* (* eps (- (* eps (* x eps)) x)) (* 0.5 x)) (/ 1.0 eps) 1.0)
   (/ (* (* 0.5 x) (* x (* eps (* eps eps)))) eps)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 9.5e-5) {
		tmp = fma(((eps * ((eps * (x * eps)) - x)) * (0.5 * x)), (1.0 / eps), 1.0);
	} else {
		tmp = ((0.5 * x) * (x * (eps * (eps * eps)))) / eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 9.5e-5)
		tmp = fma(Float64(Float64(eps * Float64(Float64(eps * Float64(x * eps)) - x)) * Float64(0.5 * x)), Float64(1.0 / eps), 1.0);
	else
		tmp = Float64(Float64(Float64(0.5 * x) * Float64(x * Float64(eps * Float64(eps * eps)))) / eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 9.5e-5], N[(N[(N[(eps * N[(N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000005e-5

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 96.6%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right)} \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon} + 1 \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon} + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + \color{blue}{x \cdot \varepsilon}\right)\right)}{\varepsilon} + 1 \]

      4. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \color{blue}{\left(0 + x \cdot \varepsilon\right)}\right)}{\varepsilon} + 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{\varepsilon \cdot \left(0 + x \cdot \varepsilon\right)}\right)}{\varepsilon} + 1 \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \varepsilon \cdot \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{0 + \color{blue}{\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      8. +-lft-identity N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \frac{\color{blue}{\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}}{\varepsilon} + 1 \]

      9. div-inv N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right) \cdot \frac{1}{\varepsilon}\right)} + 1 \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right)\right) \cdot \frac{1}{\varepsilon}} + 1 \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon \cdot \left(\left(\mathsf{neg}\left(x\right)\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)\right), \frac{1}{\varepsilon}, 1\right)} \]

   10. Applied rewrites 97.1%

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

   if 9.5000000000000005e-5 < 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

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 72.2

           \[\leadsto 0.5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Applied rewrites 72.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

       6. pow-prod-up N/A

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

       7. cube-unmult N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. inv-pow N/A

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

       11. div-inv N/A

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

       12. associate-*r* N/A

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

       13. associate-/l* N/A

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

       14. lift-*.f64 N/A

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

       15. lift-/.f64 N/A

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

       16. associate-*l* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. associate-*r/ N/A

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

       20. lower-/.f64 N/A

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

   13. Applied rewrites 81.9%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 0.5\right)}{\varepsilon}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right) - x\right)\right) \cdot \left(0.5 \cdot x\right), \frac{1}{\varepsilon}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 9.5e-5)
   (fma (* 0.5 x) (* x (* eps eps)) 1.0)
   (/ (* (* 0.5 x) (* x (* eps (* eps eps)))) eps)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 9.5e-5) {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	} else {
		tmp = ((0.5 * x) * (x * (eps * (eps * eps)))) / eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 9.5e-5)
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	else
		tmp = Float64(Float64(Float64(0.5 * x) * Float64(x * Float64(eps * Float64(eps * eps)))) / eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 9.5e-5], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000005e-5

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if 9.5000000000000005e-5 < 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

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 72.2

           \[\leadsto 0.5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Applied rewrites 72.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. associate-*r* N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

       6. pow-prod-up N/A

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

       7. cube-unmult N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. inv-pow N/A

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

       11. div-inv N/A

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

       12. associate-*r* N/A

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

       13. associate-/l* N/A

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

       14. lift-*.f64 N/A

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

       15. lift-/.f64 N/A

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

       16. associate-*l* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-/.f64 N/A

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

       19. associate-*r/ N/A

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

       20. lower-/.f64 N/A

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

   13. Applied rewrites 81.9%

       \[\leadsto \color{blue}{\frac{\left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot 0.5\right)}{\varepsilon}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 \cdot x\right) \cdot \left(x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+96}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.6)
   (fma x (fma x (fma x -0.16666666666666666 0.5) -1.0) 1.0)
   (if (<= x 1.05e+96) 0.0 (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.6) {
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else if (x <= 1.05e+96) {
		tmp = 0.0;
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.6)
		tmp = fma(x, fma(x, fma(x, -0.16666666666666666, 0.5), -1.0), 1.0);
	elseif (x <= 1.05e+96)
		tmp = 0.0;
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.6], N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.05e+96], 0.0, N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6000000000000001

   1. Initial program 62.9%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{-1 \cdot x}} \cdot 1 \]

      7. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-1 \cdot x}} \cdot 1 \]

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 77.5

         \[\leadsto e^{\color{blue}{-x}} \cdot 1 \]

   8. Applied rewrites 77.5%

      \[\leadsto \color{blue}{e^{-x} \cdot 1} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, 1\right)} \]

       3. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \color{blue}{-1}, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{-1}{6} \cdot x, -1\right)}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot x + \frac{1}{2}}, -1\right), 1\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

       8. lower-fma.f64 73.5

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)}, -1\right), 1\right) \]

   11. Applied rewrites 73.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \]

   if 1.6000000000000001 < x < 1.0500000000000001e96

   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

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right)}{\varepsilon} \]

      4. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{\varepsilon} \]

      6. metadata-eval N/A

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

      7. +-inverses N/A

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

      8. div-sub N/A

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

      9. +-inverses 52.6

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

   8. Applied rewrites 52.6%

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

   if 1.0500000000000001e96 < 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 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-out-- N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{3} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 57.2

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

   8. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+96}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 520.0)
   (fma x (fma x 0.5 -1.0) 1.0)
   (if (<= x 1.05e+96) 0.0 (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = fma(x, fma(x, 0.5, -1.0), 1.0);
	} else if (x <= 1.05e+96) {
		tmp = 0.0;
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 520.0)
		tmp = fma(x, fma(x, 0.5, -1.0), 1.0);
	elseif (x <= 1.05e+96)
		tmp = 0.0;
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 520.0], N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.05e+96], 0.0, N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 520

   1. Initial program 62.9%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{-1 \cdot x}} \cdot 1 \]

      7. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-1 \cdot x}} \cdot 1 \]

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 77.5

         \[\leadsto e^{\color{blue}{-x}} \cdot 1 \]

   8. Applied rewrites 77.5%

      \[\leadsto \color{blue}{e^{-x} \cdot 1} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 71.5

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

   11. Applied rewrites 71.5%

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

   if 520 < x < 1.0500000000000001e96

   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

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

   5. Applied rewrites 28.3%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right)}{\varepsilon} \]

      4. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{\varepsilon} \]

      6. metadata-eval N/A

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

      7. +-inverses N/A

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

      8. div-sub N/A

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

      9. +-inverses 52.6

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

   8. Applied rewrites 52.6%

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

   if 1.0500000000000001e96 < 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 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt1-in N/A

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

      7. distribute-rgt-out-- N/A

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

      8. distribute-lft-out N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{3} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. lower-fma.f64 57.2

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

   8. Applied rewrites 57.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 81.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* eps eps))))
   (if (<= x 9.5e-5) (fma (* 0.5 x) t_0 1.0) (* 0.5 (* x t_0)))))

C

double code(double x, double eps) {
	double t_0 = x * (eps * eps);
	double tmp;
	if (x <= 9.5e-5) {
		tmp = fma((0.5 * x), t_0, 1.0);
	} else {
		tmp = 0.5 * (x * t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps * eps))
	tmp = 0.0
	if (x <= 9.5e-5)
		tmp = fma(Float64(0.5 * x), t_0, 1.0);
	else
		tmp = Float64(0.5 * Float64(x * t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.5e-5], N[(N[(0.5 * x), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision], N[(0.5 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000005e-5

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 92.2

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

   8. Applied rewrites 92.2%

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

   if 9.5000000000000005e-5 < 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

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 47.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \color{blue}{\frac{0 + \varepsilon \cdot \left(\left(-x\right) + \varepsilon \cdot \left(0 + x \cdot \varepsilon\right)\right)}{\varepsilon}}, 1\right) \]

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 72.2

           \[\leadsto 0.5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

   11. Applied rewrites 72.2%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 64.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{if}\;x \leq 520:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.5 -1.0) 1.0)))
   (if (<= x 520.0) t_0 (if (<= x 5e+154) 0.0 t_0))))

C

double code(double x, double eps) {
	double t_0 = fma(x, fma(x, 0.5, -1.0), 1.0);
	double tmp;
	if (x <= 520.0) {
		tmp = t_0;
	} else if (x <= 5e+154) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(x, fma(x, 0.5, -1.0), 1.0)
	tmp = 0.0
	if (x <= 520.0)
		tmp = t_0;
	elseif (x <= 5e+154)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, 520.0], t$95$0, If[LessEqual[x, 5e+154], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 520 or 5.00000000000000004e154 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{-1 \cdot x}} \cdot 1 \]

      7. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-1 \cdot x}} \cdot 1 \]

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 72.8

         \[\leadsto e^{\color{blue}{-x}} \cdot 1 \]

   8. Applied rewrites 72.8%

      \[\leadsto \color{blue}{e^{-x} \cdot 1} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. lower-fma.f64 69.8

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

   11. Applied rewrites 69.8%

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

   if 520 < x < 5.00000000000000004e154

   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

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right)}{\varepsilon} \]

      4. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{\varepsilon} \]

      6. metadata-eval N/A

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

      7. +-inverses N/A

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

      8. div-sub N/A

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

      9. +-inverses 47.0

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

   8. Applied rewrites 47.0%

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

Alternative 13: 57.7% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 1.0) (- 1.0 x) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} else {
		tmp = 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 <= 1.0d0) then
        tmp = 1.0d0 - x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 62.9%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{-1 \cdot x}} \cdot 1 \]

      7. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{-1 \cdot x}} \cdot 1 \]

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 77.5

         \[\leadsto e^{\color{blue}{-x}} \cdot 1 \]

   8. Applied rewrites 77.5%

      \[\leadsto \color{blue}{e^{-x} \cdot 1} \]

   9. Taylor expanded in x around 0

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

       1. neg-mul-1 N/A

          \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

       2. unsub-neg N/A

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

       3. lower--.f64 60.0

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

   11. Applied rewrites 60.0%

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

   if 1 < 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

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

   5. Applied rewrites 47.0%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right)}{\varepsilon} \]

      4. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{\varepsilon} \]

      6. metadata-eval N/A

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

      7. +-inverses N/A

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

      8. div-sub N/A

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

      9. +-inverses 45.0

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

   8. Applied rewrites 45.0%

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

Alternative 14: 57.7% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 520.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 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 <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 520.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 520

   1. Initial program 62.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 59.9%

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

   if 520 < 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

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

   5. Applied rewrites 47.0%

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

   6. Taylor expanded in eps around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right)}{\varepsilon} \]

      4. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

      5. mul0-rgt N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{\varepsilon} \]

      6. metadata-eval N/A

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

      7. +-inverses N/A

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

      8. div-sub N/A

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

      9. +-inverses 45.0

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

   8. Applied rewrites 45.0%

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

Alternative 15: 15.9% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 79.7%

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

6. Taylor expanded in eps around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{0}\right)\right)}{\varepsilon} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \left(x \cdot \color{blue}{0}\right)}{\varepsilon} \]

   5. mul0-rgt N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{0}}{\varepsilon} \]

   6. metadata-eval N/A

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

   7. +-inverses N/A

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

   8. div-sub N/A

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

   9. +-inverses 13.8

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

8. Applied rewrites 13.8%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))