NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.8%

Time: 17.2s

Alternatives: 12

Speedup: 9.7×

• 37.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.92485100273742e+167
  2. eps = -1.524793875802782e-78

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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps)))))
   (if (<=
        (+
         (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ -1.0 eps))))
         (* t_0 (+ 1.0 (/ -1.0 eps))))
        0.0)
     (* 0.5 (* (exp (- 0.0 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 * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (exp((0.0 - 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 * ((-1.0d0) + eps)))) + (t_0 * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = 0.5d0 * (exp((0.0d0 - 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 * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (Math.exp((0.0 - 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 * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = 0.5 * (math.exp((0.0 - 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(-1.0 + eps)))) + Float64(t_0 * Float64(1.0 + Float64(-1.0 / eps)))) <= 0.0)
		tmp = Float64(0.5 * Float64(exp(Float64(0.0 - 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 * (-1.0 + eps)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 0.0)
		tmp = 0.5 * (exp((0.0 - 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[(-1.0 + eps), $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[N[(0.0 - x), $MachinePrecision]], $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 43.9%

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

   3. Taylor expanded in eps around 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. *-lowering-*.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. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-sub0 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{0 - x}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{0 - x}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{0 - x} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{0 - x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]

   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. *-lowering-*.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. +-lowering-+.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. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;0.5 \cdot \left(e^{0 - x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \varepsilon, -1\right), x\right)}{\varepsilon}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ -1.0 eps))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   (* 0.5 (* (exp (- 0.0 x)) (+ x (+ x 2.0))))
   (fma 0.5 (/ (fma (* eps eps) (* x (fma x eps -1.0)) x) eps) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = 0.5 * (exp((0.0 - x)) * (x + (x + 2.0)));
	} else {
		tmp = fma(0.5, (fma((eps * eps), (x * fma(x, eps, -1.0)), x) / 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = Float64(0.5 * Float64(exp(Float64(0.0 - x)) * Float64(x + Float64(x + 2.0))));
	else
		tmp = fma(0.5, Float64(fma(Float64(eps * eps), Float64(x * fma(x, eps, -1.0)), x) / 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[(-1.0 + eps), $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[(0.5 * N[(N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(eps * eps), $MachinePrecision] * N[(x * N[(x * eps + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / 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))))) < 4

   1. Initial program 62.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      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. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-sub0 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{0 - x}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{0 - x}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{0 - x} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{0 - x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]

   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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. +-commutative N/A

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

       10. metadata-eval N/A

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

       11. sub-neg N/A

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

       12. *-lowering-*.f64 N/A

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

       13. sub-neg N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

       16. accelerator-lowering-fma.f64 87.5

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

   11. Simplified 87.5%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;0.5 \cdot \left(e^{0 - x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \varepsilon, -1\right), x\right)}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \varepsilon, -1\right), x\right)}{\varepsilon}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ -1.0 eps))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   (exp (- 0.0 x))
   (fma 0.5 (/ (fma (* eps eps) (* x (fma x eps -1.0)) x) eps) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = exp((0.0 - x));
	} else {
		tmp = fma(0.5, (fma((eps * eps), (x * fma(x, eps, -1.0)), x) / 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = exp(Float64(0.0 - x));
	else
		tmp = fma(0.5, Float64(fma(Float64(eps * eps), Float64(x * fma(x, eps, -1.0)), x) / 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[(-1.0 + eps), $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[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[(N[(eps * eps), $MachinePrecision] * N[(x * N[(x * eps + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / 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))))) < 4

   1. Initial program 62.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. *-lowering-*.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. +-lowering-+.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. Simplified 98.6%

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

   6. Taylor expanded in eps around 0

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

      1. exp-lowering-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. neg-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - x}} \]

      4. --lowering--.f64 98.3

         \[\leadsto e^{\color{blue}{0 - x}} \]

   8. Simplified 98.3%

      \[\leadsto \color{blue}{e^{0 - x}} \]

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. +-commutative N/A

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

       10. metadata-eval N/A

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

       11. sub-neg N/A

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

       12. *-lowering-*.f64 N/A

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

       13. sub-neg N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

       16. accelerator-lowering-fma.f64 87.5

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

   11. Simplified 87.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, x \cdot \mathsf{fma}\left(x, \varepsilon, -1\right), x\right)}{\varepsilon}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = fma(0.5, (x * (eps * (x * eps))), 1.0);
	} else {
		tmp = fma(0.5, (fma((eps * eps), (x * fma(x, eps, -1.0)), x) / 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = fma(0.5, Float64(x * Float64(eps * Float64(x * eps))), 1.0);
	else
		tmp = fma(0.5, Float64(fma(Float64(eps * eps), Float64(x * fma(x, eps, -1.0)), x) / 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[(-1.0 + eps), $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[(0.5 * N[(x * N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(N[(N[(eps * eps), $MachinePrecision] * N[(x * N[(x * eps + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / 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))))) < 4

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 65.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.0

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

   8. Simplified 66.0%

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

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 72.1

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

   10. Applied egg-rr 72.1%

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

   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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. +-commutative N/A

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

       10. metadata-eval N/A

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

       11. sub-neg N/A

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

       12. *-lowering-*.f64 N/A

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

       13. sub-neg N/A

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

       14. *-commutative N/A

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

       15. metadata-eval N/A

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

       16. accelerator-lowering-fma.f64 87.5

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

   11. Simplified 87.5%

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

4. Final simplification 79.0%

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

Alternative 5: 80.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = fma(0.5, (x * (eps * (x * eps))), 1.0);
	} else {
		tmp = fma(0.5, (x * (fma((eps * eps), fma(x, eps, -1.0), 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = fma(0.5, Float64(x * Float64(eps * Float64(x * eps))), 1.0);
	else
		tmp = fma(0.5, Float64(x * Float64(fma(Float64(eps * eps), fma(x, eps, -1.0), 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[(-1.0 + eps), $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[(0.5 * N[(x * N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(x * eps + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $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))))) < 4

   1. Initial program 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 65.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 66.0

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

   8. Simplified 66.0%

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

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 72.1

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

   10. Applied egg-rr 72.1%

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

   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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. sub-neg N/A

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. accelerator-lowering-fma.f64 86.0

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

   11. Simplified 86.0%

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

4. Final simplification 78.3%

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

Alternative 6: 78.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = fma(((eps * eps) * (0.5 * x)), x, 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(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.0)
		tmp = 1.0;
	else
		tmp = fma(Float64(Float64(eps * eps) * Float64(0.5 * x)), x, 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[(-1.0 + eps), $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], 1.0, N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] * x + 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 62.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.2%

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

   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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

      1. associate-*r* N/A

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 84.4

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

   10. Applied egg-rr 84.4%

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

4. Final simplification 77.6%

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

Alternative 7: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = 1.0;
	else
		tmp = Float64(eps * Float64(x * fma(Float64(0.5 * eps), x, -0.5)));
	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[(-1.0 + eps), $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], 1.0, N[(eps * N[(x * N[(N[(0.5 * eps), $MachinePrecision] * x + -0.5), $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 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.0%

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

   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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

   9. Taylor expanded in eps around inf

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

       1. distribute-lft-in N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. *-rgt-identity N/A

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

       9. associate-*r/ N/A

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

       10. rgt-mult-inverse N/A

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

       11. *-rgt-identity N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. distribute-rgt-out N/A

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

       16. +-commutative N/A

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

       17. *-commutative N/A

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

       18. associate-*r* N/A

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

       19. *-commutative N/A

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

   11. Simplified 65.0%

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

4. Final simplification 68.9%

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

Alternative 8: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * ((-1.0d0) + eps)))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 4.0d0) then
        tmp = 1.0d0
    else
        tmp = eps * (eps * (0.5d0 * (x * x)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (-1.0 + eps)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0:
		tmp = 1.0
	else:
		tmp = eps * (eps * (0.5 * (x * x)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0)
		tmp = 1.0;
	else
		tmp = eps * (eps * (0.5 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 + eps), $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], 1.0, N[(eps * N[(eps * N[(0.5 * N[(x * x), $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 62.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.0%

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

   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. Simplified 84.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 84.4

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

   8. Simplified 84.4%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 64.7

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

   11. Simplified 64.7%

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

4. Final simplification 68.8%

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

Alternative 9: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.5d0 * (exp((x * ((-1.0d0) + eps))) + exp((x * ((-1.0d0) - eps))))
end function

Java

public static double code(double x, double eps) {
	return 0.5 * (Math.exp((x * (-1.0 + eps))) + Math.exp((x * (-1.0 - eps))));
}

Python

def code(x, eps):
	return 0.5 * (math.exp((x * (-1.0 + eps))) + math.exp((x * (-1.0 - eps))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (exp((x * (-1.0 + eps))) + exp((x * (-1.0 - eps))));
end

Wolfram

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

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in eps around 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. *-lowering-*.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. +-lowering-+.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. Simplified 99.2%

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

Alternative 10: 76.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.4e-7)
   (fma 0.5 (* x (* eps (* x eps))) 1.0)
   (* eps (* eps (* 0.5 (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.4e-7) {
		tmp = fma(0.5, (x * (eps * (x * eps))), 1.0);
	} else {
		tmp = eps * (eps * (0.5 * (x * x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.4e-7)
		tmp = fma(0.5, Float64(x * Float64(eps * Float64(x * eps))), 1.0);
	else
		tmp = Float64(eps * Float64(eps * Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.4e-7], N[(0.5 * N[(x * N[(eps * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(eps * N[(eps * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999979e-7

   1. Initial program 69.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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. Simplified 88.6%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 88.8

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

   8. Simplified 88.8%

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

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 86.1

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

   10. Applied egg-rr 86.1%

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

   if 2.39999999999999979e-7 < 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. Simplified 41.4%

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

   6. Taylor expanded in eps around inf

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

      1. +-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. mul0-lft N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 41.9

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

   8. Simplified 41.9%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 49.1

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

   11. Simplified 49.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \left(\varepsilon \cdot \left(x \cdot \varepsilon\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0))

C

double code(double x, double eps) {
	return fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in eps around 0

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

   1. *-lowering-*.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. distribute-rgt1-in N/A

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

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

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

   7. *-commutative N/A

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

   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. *-lowering-*.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. neg-sub0 N/A

       \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{0 - x}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(e^{\color{blue}{0 - x}} \cdot \left(\left(\left(x + 1\right) - -1\right) + x\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(e^{0 - x} \cdot \color{blue}{\left(\left(\left(x + 1\right) - -1\right) + x\right)}\right) \]

5. Simplified 55.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{0 - x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]

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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-fma.f64 50.0

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

8. Simplified 50.0%

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

Alternative 12: 44.8% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in x around 0

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

5. Simplified 41.3%

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

Reproduce

herbie shell --seed 2024195 
(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))