NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.9% → 99.1%

Time: 11.5s

Alternatives: 10

Speedup: 8.3×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.2367810732531882e+72
  2. eps = 5.2087638729728726e-288

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% 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: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.000000005377993:\\ \;\;\;\;\left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(-1 + \varepsilon\right) \cdot \left(-1 + \varepsilon\right)\right) \cdot 0.5, x, \left(-2 - \varepsilon\right) + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps) x)) (+ (/ 1.0 eps) 1.0))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      2.000000005377993)
   (* (* (+ (+ 2.0 x) x) (exp (- x))) 0.5)
   (*
    (fma
     (fma
      (* (fma (+ eps 1.0) (+ eps 1.0) (* (+ -1.0 eps) (+ -1.0 eps))) 0.5)
      x
      (+ (- -2.0 eps) eps))
     x
     2.0)
    0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.000000005377993) {
		tmp = (((2.0 + x) + x) * exp(-x)) * 0.5;
	} else {
		tmp = fma(fma((fma((eps + 1.0), (eps + 1.0), ((-1.0 + eps) * (-1.0 + eps))) * 0.5), x, ((-2.0 - eps) + eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 2.000000005377993)
		tmp = Float64(Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(fma(fma(Float64(fma(Float64(eps + 1.0), Float64(eps + 1.0), Float64(Float64(-1.0 + eps) * Float64(-1.0 + eps))) * 0.5), x, Float64(Float64(-2.0 - eps) + eps)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.000000005377993], N[(N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision] + N[(N[(-1.0 + eps), $MachinePrecision] * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + N[(N[(-2.0 - eps), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $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))))) < 2.0000000053779932

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if 2.0000000053779932 < (-.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 99.2%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.000000005377993:\\ \;\;\;\;\left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(-1 + \varepsilon\right) \cdot \left(-1 + \varepsilon\right)\right) \cdot 0.5, x, \left(-2 - \varepsilon\right) + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.000000005377993:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(-1 + \varepsilon\right) \cdot \left(-1 + \varepsilon\right)\right) \cdot 0.5, x, \left(-2 - \varepsilon\right) + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps) x)) (+ (/ 1.0 eps) 1.0))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      2.000000005377993)
   (exp (- x))
   (*
    (fma
     (fma
      (* (fma (+ eps 1.0) (+ eps 1.0) (* (+ -1.0 eps) (+ -1.0 eps))) 0.5)
      x
      (+ (- -2.0 eps) eps))
     x
     2.0)
    0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.000000005377993) {
		tmp = exp(-x);
	} else {
		tmp = fma(fma((fma((eps + 1.0), (eps + 1.0), ((-1.0 + eps) * (-1.0 + eps))) * 0.5), x, ((-2.0 - eps) + eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 2.000000005377993)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(fma(fma(Float64(fma(Float64(eps + 1.0), Float64(eps + 1.0), Float64(Float64(-1.0 + eps) * Float64(-1.0 + eps))) * 0.5), x, Float64(Float64(-2.0 - eps) + eps)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.000000005377993], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(N[(N[(N[(eps + 1.0), $MachinePrecision] * N[(eps + 1.0), $MachinePrecision] + N[(N[(-1.0 + eps), $MachinePrecision] * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + N[(N[(-2.0 - eps), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $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))))) < 2.0000000053779932

   1. Initial program 50.3%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 98.4%

      \[\leadsto e^{-x} \]

   if 2.0000000053779932 < (-.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 99.2%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.000000005377993:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon + 1, \varepsilon + 1, \left(-1 + \varepsilon\right) \cdot \left(-1 + \varepsilon\right)\right) \cdot 0.5, x, \left(-2 - \varepsilon\right) + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps) x)) (+ (/ 1.0 eps) 1.0))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      5.0)
   1.0
   (* (* x x) (* (* eps eps) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * ((eps * eps) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (((exp((((-1.0d0) + eps) * x)) * ((1.0d0 / eps) + 1.0d0)) - (exp((((-1.0d0) - eps) * x)) * ((1.0d0 / eps) - 1.0d0))) <= 5.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * ((eps * eps) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (((Math.exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (Math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * ((eps * eps) * 0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if ((math.exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 5.0:
		tmp = 1.0
	else:
		tmp = (x * x) * ((eps * eps) * 0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 5.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * eps) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 5.0)
		tmp = 1.0;
	else
		tmp = (x * x) * ((eps * eps) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5.0], 1.0, N[(N[(x * x), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.5), $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))))) < 5

   1. Initial program 50.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.6%

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

   if 5 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \mathsf{fma}\left(-1 - \varepsilon, \frac{1 + \varepsilon}{\varepsilon} - \left(1 + \varepsilon\right), \left(\frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot \left(\varepsilon - 1\right)\right), x, \mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot 0.5\right), x, 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 68.7%

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

4. Final simplification 73.9%

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

Alternative 5: 78.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 45:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \varepsilon, \varepsilon, -0.5\right) \cdot x, x, 1\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;0 \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 45.0)
   (fma (* (fma (* 0.5 eps) eps -0.5) x) x 1.0)
   (if (<= x 1.5e+80) (* 0.0 eps) (* (* x x) (* (* eps eps) 0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 45.0) {
		tmp = fma((fma((0.5 * eps), eps, -0.5) * x), x, 1.0);
	} else if (x <= 1.5e+80) {
		tmp = 0.0 * eps;
	} else {
		tmp = (x * x) * ((eps * eps) * 0.5);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 45.0)
		tmp = fma(Float64(fma(Float64(0.5 * eps), eps, -0.5) * x), x, 1.0);
	elseif (x <= 1.5e+80)
		tmp = Float64(0.0 * eps);
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * eps) * 0.5));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 45.0], N[(N[(N[(N[(0.5 * eps), $MachinePrecision] * eps + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[x, 1.5e+80], N[(0.0 * eps), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 45

   1. Initial program 60.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \mathsf{fma}\left(-1 - \varepsilon, \frac{1 + \varepsilon}{\varepsilon} - \left(1 + \varepsilon\right), \left(\frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot \left(\varepsilon - 1\right)\right), x, \mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot 0.5\right), x, 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.7%

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

   if 45 < x < 1.49999999999999993e80

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 11.0%

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

   9. Applied rewrites 46.5%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 59.1%

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

   if 1.49999999999999993e80 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \mathsf{fma}\left(-1 - \varepsilon, \frac{1 + \varepsilon}{\varepsilon} - \left(1 + \varepsilon\right), \left(\frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot \left(\varepsilon - 1\right)\right), x, \mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot 0.5\right), x, 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 51.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 64.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 45:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \varepsilon, \varepsilon, -0.5\right) \cdot x, x, 1\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;0 \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 780:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot 0.5, x, 1\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;0 \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 780.0)
   (fma (* (* (* eps eps) x) 0.5) x 1.0)
   (if (<= x 1.5e+80) (* 0.0 eps) (* (* x x) (* (* eps eps) 0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 780.0) {
		tmp = fma((((eps * eps) * x) * 0.5), x, 1.0);
	} else if (x <= 1.5e+80) {
		tmp = 0.0 * eps;
	} else {
		tmp = (x * x) * ((eps * eps) * 0.5);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 780.0)
		tmp = fma(Float64(Float64(Float64(eps * eps) * x) * 0.5), x, 1.0);
	elseif (x <= 1.5e+80)
		tmp = Float64(0.0 * eps);
	else
		tmp = Float64(Float64(x * x) * Float64(Float64(eps * eps) * 0.5));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 780.0], N[(N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[x, 1.5e+80], N[(0.0 * eps), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 780

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \mathsf{fma}\left(-1 - \varepsilon, \frac{1 + \varepsilon}{\varepsilon} - \left(1 + \varepsilon\right), \left(\frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot \left(\varepsilon - 1\right)\right), x, \mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot 0.5\right), x, 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 89.2%

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

   if 780 < x < 1.49999999999999993e80

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 2.9%

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

   7. Applied rewrites 11.1%

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

   9. Applied rewrites 48.4%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 61.5%

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

   if 1.49999999999999993e80 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \mathsf{fma}\left(-1 - \varepsilon, \frac{1 + \varepsilon}{\varepsilon} - \left(1 + \varepsilon\right), \left(\frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot \left(\varepsilon - 1\right)\right), x, \mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \frac{\varepsilon - 1}{\varepsilon} + \left(\varepsilon - 1\right)\right) \cdot 0.5\right), x, 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 51.2%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 64.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 780:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot 0.5, x, 1\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+80}:\\ \;\;\;\;0 \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 10^{+215}:\\ \;\;\;\;0 \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.0)
   (* (fma -2.0 x 2.0) 0.5)
   (if (<= x 1e+215) (* 0.0 eps) (* (fma (fma x 0.5 -1.0) x 1.0) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else if (x <= 1e+215) {
		tmp = 0.0 * eps;
	} else {
		tmp = fma(fma(x, 0.5, -1.0), x, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	elseif (x <= 1e+215)
		tmp = Float64(0.0 * eps);
	else
		tmp = Float64(fma(fma(x, 0.5, -1.0), x, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.0], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1e+215], N[(0.0 * eps), $MachinePrecision], N[(N[(N[(x * 0.5 + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 60.4%

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

   3. Taylor expanded in eps around 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 62.9%

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

   if 1 < x < 9.99999999999999907e214

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 6.1%

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

   9. Applied rewrites 27.4%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 54.6%

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

   if 9.99999999999999907e214 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 32.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 69.2%

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

Alternative 8: 57.7% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 60.4%

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

   3. Taylor expanded in eps around 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.6%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 62.9%

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

   if 1 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 5.2%

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

   9. Applied rewrites 23.9%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 49.4%

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

Alternative 9: 57.7% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0 \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 550.0) 1.0 (* 0.0 eps)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 * eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 550.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0 * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0 * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = Float64(0.0 * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0 * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, N[(0.0 * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 60.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.6%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.0%

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

   7. Applied rewrites 5.1%

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

   9. Applied rewrites 24.2%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 50.0%

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

Alternative 10: 44.0% 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 70.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 47.0%

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

Reproduce

herbie shell --seed 2024235 
(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))