NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.8%

Time: 14.4s

Alternatives: 16

Speedup: 9.7×

• 37.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.307293209348714e+272
  2. eps = -2.328928165730991e-151

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: 73.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 31.1%

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

   3. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. 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. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.05:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \varepsilon - x} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\varepsilon + 1\right) \cdot \mathsf{fma}\left(0.5, \varepsilon, 0.5\right), -1 - \varepsilon\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      2.05)
   (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
   (*
    0.5
    (+
     (exp (- (* x eps) x))
     (fma x (fma x (* (+ eps 1.0) (fma 0.5 eps 0.5)) (- -1.0 eps)) 1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 2.05) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (exp(((x * eps) - x)) + fma(x, fma(x, ((eps + 1.0) * fma(0.5, eps, 0.5)), (-1.0 - eps)), 1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.05)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) * Float64(x + Float64(x + 2.0))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(Float64(x * eps) - x)) + fma(x, fma(x, Float64(Float64(eps + 1.0) * fma(0.5, eps, 0.5)), Float64(-1.0 - eps)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.05], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[(x * N[(x * N[(N[(eps + 1.0), $MachinePrecision] * N[(0.5 * eps + 0.5), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] + 1.0), $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))))) < 2.0499999999999998

   1. Initial program 51.9%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. 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. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if 2.0499999999999998 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 89.1%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.05:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \varepsilon - x} + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\varepsilon + 1\right) \cdot \mathsf{fma}\left(0.5, \varepsilon, 0.5\right), -1 - \varepsilon\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.05:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      2.05)
   (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
   (fma (* 0.5 x) (* x (* eps eps)) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 2.05) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.05)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) * Float64(x + Float64(x + 2.0))));
	else
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.05], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * 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))))) < 2.0499999999999998

   1. Initial program 51.9%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. 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. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if 2.0499999999999998 < (-.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 84.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.05:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.05:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      2.05)
   (exp (- x))
   (fma (* 0.5 x) (* x (* eps eps)) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 2.05) {
		tmp = exp(-x);
	} else {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.05)
		tmp = exp(Float64(-x));
	else
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.05], N[Exp[(-x)], $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * 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))))) < 2.0499999999999998

   1. Initial program 51.9%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in eps around 0

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

      1. neg-mul-1 N/A

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

      2. lower-exp.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 2.0499999999999998 < (-.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 84.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2.05:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. neg-mul-1 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 6: 74.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * (0.5 * (eps * eps));
	}
	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 * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 5.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * x) * (0.5d0 * (eps * eps))
    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 * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = (x * x) * (0.5 * (eps * eps));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 5.0)
		tmp = 1.0;
	else
		tmp = (x * x) * (0.5 * (eps * eps));
	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[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5.0], 1.0, N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps * eps), $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))))) < 5

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.1%

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

   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 85.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 75.2

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

   8. Applied rewrites 75.2%

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

4. Final simplification 77.5%

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

Alternative 7: 71.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps * (x * (x * eps)));
	}
	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 * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 5.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (eps * (x * (x * eps)))
    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 * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps * (x * (x * eps)));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 5.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (eps * (x * (x * eps)));
	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[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5.0], 1.0, N[(0.5 * N[(eps * N[(x * 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))))) < 5

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.1%

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

   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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)\right)} \]

   5. Applied rewrites 12.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 50.9

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \color{blue}{\mathsf{fma}\left(x, -0.6666666666666666, 1\right)}\right)\right) \]

   8. Applied rewrites 50.9%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.5 \cdot \mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)\right)} \]

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 67.7

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

   11. Applied rewrites 67.7%

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

4. Final simplification 74.4%

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

Alternative 8: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-+.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 9: 83.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - \varepsilon\\ \mathbf{if}\;x \leq -9 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right), \frac{\varepsilon + -1}{\varepsilon} + \left(\varepsilon + -1\right)\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 0.085:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 - \varepsilon, t\_0, \left(1 - \varepsilon\right) \cdot t\_0\right) \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) eps)))
   (if (<= x -9e-233)
     (fma
      x
      (*
       0.5
       (+
        (+ (- -1.0 eps) (/ (+ eps 1.0) eps))
        (fma
         x
         (* eps (* eps (fma x -0.6666666666666666 1.0)))
         (+ (/ (+ eps -1.0) eps) (+ eps -1.0)))))
      1.0)
     (if (<= x 5.8e-265)
       (fma x (* (+ eps -1.0) (fma (* x 0.25) (+ eps -1.0) 0.5)) 1.0)
       (if (<= x 0.085)
         (fma (* 0.5 x) (* x (* eps eps)) 1.0)
         (* (fma (- -1.0 eps) t_0 (* (- 1.0 eps) t_0)) (* 0.25 (* x x))))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - eps;
	double tmp;
	if (x <= -9e-233) {
		tmp = fma(x, (0.5 * (((-1.0 - eps) + ((eps + 1.0) / eps)) + fma(x, (eps * (eps * fma(x, -0.6666666666666666, 1.0))), (((eps + -1.0) / eps) + (eps + -1.0))))), 1.0);
	} else if (x <= 5.8e-265) {
		tmp = fma(x, ((eps + -1.0) * fma((x * 0.25), (eps + -1.0), 0.5)), 1.0);
	} else if (x <= 0.085) {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	} else {
		tmp = fma((-1.0 - eps), t_0, ((1.0 - eps) * t_0)) * (0.25 * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - eps)
	tmp = 0.0
	if (x <= -9e-233)
		tmp = fma(x, Float64(0.5 * Float64(Float64(Float64(-1.0 - eps) + Float64(Float64(eps + 1.0) / eps)) + fma(x, Float64(eps * Float64(eps * fma(x, -0.6666666666666666, 1.0))), Float64(Float64(Float64(eps + -1.0) / eps) + Float64(eps + -1.0))))), 1.0);
	elseif (x <= 5.8e-265)
		tmp = fma(x, Float64(Float64(eps + -1.0) * fma(Float64(x * 0.25), Float64(eps + -1.0), 0.5)), 1.0);
	elseif (x <= 0.085)
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	else
		tmp = Float64(fma(Float64(-1.0 - eps), t_0, Float64(Float64(1.0 - eps) * t_0)) * Float64(0.25 * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - eps), $MachinePrecision]}, If[LessEqual[x, -9e-233], N[(x * N[(0.5 * N[(N[(N[(-1.0 - eps), $MachinePrecision] + N[(N[(eps + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * N[(eps * N[(x * -0.6666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(eps + -1.0), $MachinePrecision] / eps), $MachinePrecision] + N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5.8e-265], N[(x * N[(N[(eps + -1.0), $MachinePrecision] * N[(N[(x * 0.25), $MachinePrecision] * N[(eps + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 0.085], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(-1.0 - eps), $MachinePrecision] * t$95$0 + N[(N[(1.0 - eps), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.0000000000000004e-233

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)\right)} \]

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in eps around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 92.9

         \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, -0.6666666666666666, 1\right)}\right), \left(-1 + \varepsilon\right) + \frac{-1 + \varepsilon}{\varepsilon}\right)\right), 1\right) \]

   8. Applied rewrites 92.9%

      \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}, \left(-1 + \varepsilon\right) + \frac{-1 + \varepsilon}{\varepsilon}\right)\right), 1\right) \]

   if -9.0000000000000004e-233 < x < 5.7999999999999995e-265

   1. Initial program 65.6%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right) + \frac{1}{2} \cdot \left(\varepsilon - 1\right), 1\right)} \]

   11. Applied rewrites 98.1%

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

   if 5.7999999999999995e-265 < x < 0.0850000000000000061

   1. Initial program 51.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.0850000000000000061 < 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 42.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 - \varepsilon, \frac{1}{\varepsilon} - \varepsilon, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)\right) \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 \cdot \left(\left(\left(-1 - \varepsilon\right) + \frac{\varepsilon + 1}{\varepsilon}\right) + \mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right), \frac{\varepsilon + -1}{\varepsilon} + \left(\varepsilon + -1\right)\right)\right), 1\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 0.085:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 - \varepsilon, \frac{1}{\varepsilon} - \varepsilon, \left(1 - \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - \varepsilon\right)\right) \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - \varepsilon\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 0.085:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 - \varepsilon, t\_0, \left(1 - \varepsilon\right) \cdot t\_0\right) \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) eps)))
   (if (<= x 5.8e-265)
     (fma x (* (+ eps -1.0) (fma (* x 0.25) (+ eps -1.0) 0.5)) 1.0)
     (if (<= x 0.085)
       (fma (* 0.5 x) (* x (* eps eps)) 1.0)
       (* (fma (- -1.0 eps) t_0 (* (- 1.0 eps) t_0)) (* 0.25 (* x x)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - eps;
	double tmp;
	if (x <= 5.8e-265) {
		tmp = fma(x, ((eps + -1.0) * fma((x * 0.25), (eps + -1.0), 0.5)), 1.0);
	} else if (x <= 0.085) {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	} else {
		tmp = fma((-1.0 - eps), t_0, ((1.0 - eps) * t_0)) * (0.25 * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - eps)
	tmp = 0.0
	if (x <= 5.8e-265)
		tmp = fma(x, Float64(Float64(eps + -1.0) * fma(Float64(x * 0.25), Float64(eps + -1.0), 0.5)), 1.0);
	elseif (x <= 0.085)
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	else
		tmp = Float64(fma(Float64(-1.0 - eps), t_0, Float64(Float64(1.0 - eps) * t_0)) * Float64(0.25 * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - eps), $MachinePrecision]}, If[LessEqual[x, 5.8e-265], N[(x * N[(N[(eps + -1.0), $MachinePrecision] * N[(N[(x * 0.25), $MachinePrecision] * N[(eps + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 0.085], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(-1.0 - eps), $MachinePrecision] * t$95$0 + N[(N[(1.0 - eps), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7999999999999995e-265

   1. Initial program 67.4%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right) + \frac{1}{2} \cdot \left(\varepsilon - 1\right), 1\right)} \]

   11. Applied rewrites 90.5%

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

   if 5.7999999999999995e-265 < x < 0.0850000000000000061

   1. Initial program 51.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   if 0.0850000000000000061 < 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 42.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 - \varepsilon, \frac{1}{\varepsilon} - \varepsilon, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + \frac{-1}{\varepsilon}\right)\right) \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 0.085:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1 - \varepsilon, \frac{1}{\varepsilon} - \varepsilon, \left(1 - \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - \varepsilon\right)\right) \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \mathsf{fma}\left(x \cdot 0.08333333333333333, \varepsilon + -1, 0.25\right), \mathsf{fma}\left(0.5, \varepsilon, -0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5.8e-265)
   (fma x (* (+ eps -1.0) (fma (* x 0.25) (+ eps -1.0) 0.5)) 1.0)
   (if (<= x 3.6e+42)
     (fma
      x
      (fma
       x
       (*
        (* (- 1.0 eps) (- 1.0 eps))
        (fma (* x 0.08333333333333333) (+ eps -1.0) 0.25))
       (fma 0.5 eps -0.5))
      1.0)
     (* (* x x) (* 0.5 (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5.8e-265) {
		tmp = fma(x, ((eps + -1.0) * fma((x * 0.25), (eps + -1.0), 0.5)), 1.0);
	} else if (x <= 3.6e+42) {
		tmp = fma(x, fma(x, (((1.0 - eps) * (1.0 - eps)) * fma((x * 0.08333333333333333), (eps + -1.0), 0.25)), fma(0.5, eps, -0.5)), 1.0);
	} else {
		tmp = (x * x) * (0.5 * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5.8e-265)
		tmp = fma(x, Float64(Float64(eps + -1.0) * fma(Float64(x * 0.25), Float64(eps + -1.0), 0.5)), 1.0);
	elseif (x <= 3.6e+42)
		tmp = fma(x, fma(x, Float64(Float64(Float64(1.0 - eps) * Float64(1.0 - eps)) * fma(Float64(x * 0.08333333333333333), Float64(eps + -1.0), 0.25)), fma(0.5, eps, -0.5)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5.8e-265], N[(x * N[(N[(eps + -1.0), $MachinePrecision] * N[(N[(x * 0.25), $MachinePrecision] * N[(eps + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 3.6e+42], N[(x * N[(x * N[(N[(N[(1.0 - eps), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.08333333333333333), $MachinePrecision] * N[(eps + -1.0), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] + N[(0.5 * eps + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7999999999999995e-265

   1. Initial program 67.4%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right) + \frac{1}{2} \cdot \left(\varepsilon - 1\right), 1\right)} \]

   11. Applied rewrites 90.5%

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

   if 5.7999999999999995e-265 < x < 3.6000000000000001e42

   1. Initial program 60.2%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{12} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{12} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot \left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{12} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right), 1\right)} \]

   11. Applied rewrites 72.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \mathsf{fma}\left(x \cdot 0.08333333333333333, \varepsilon + -1, 0.25\right), \mathsf{fma}\left(0.5, \varepsilon, -0.5\right)\right), 1\right)} \]

   if 3.6000000000000001e42 < 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 49.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 64.2

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

   8. Applied rewrites 64.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right) \cdot \mathsf{fma}\left(x \cdot 0.08333333333333333, \varepsilon + -1, 0.25\right), \mathsf{fma}\left(0.5, \varepsilon, -0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5.8e-265)
   (fma x (* (+ eps -1.0) (fma (* x 0.25) (+ eps -1.0) 0.5)) 1.0)
   (if (<= x 2.6)
     (fma (* 0.5 x) (* x (* eps eps)) 1.0)
     (* (* x x) (* 0.5 (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5.8e-265) {
		tmp = fma(x, ((eps + -1.0) * fma((x * 0.25), (eps + -1.0), 0.5)), 1.0);
	} else if (x <= 2.6) {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	} else {
		tmp = (x * x) * (0.5 * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5.8e-265)
		tmp = fma(x, Float64(Float64(eps + -1.0) * fma(Float64(x * 0.25), Float64(eps + -1.0), 0.5)), 1.0);
	elseif (x <= 2.6)
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5.8e-265], N[(x * N[(N[(eps + -1.0), $MachinePrecision] * N[(N[(x * 0.25), $MachinePrecision] * N[(eps + -1.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 2.6], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.7999999999999995e-265

   1. Initial program 67.4%

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

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{4} \cdot {\left(\varepsilon - 1\right)}^{2}\right) + \frac{1}{2} \cdot \left(\varepsilon - 1\right), 1\right)} \]

   11. Applied rewrites 90.5%

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

   if 5.7999999999999995e-265 < x < 2.60000000000000009

   1. Initial program 52.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 96.1%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 96.1

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

   8. Applied rewrites 96.1%

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

   if 2.60000000000000009 < 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 42.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.6

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

   8. Applied rewrites 57.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-265}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\varepsilon + -1\right) \cdot \mathsf{fma}\left(x \cdot 0.25, \varepsilon + -1, 0.5\right), 1\right)\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.6)
   (fma (* 0.5 x) (* x (* eps eps)) 1.0)
   (* (* x x) (* 0.5 (* eps eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.6) {
		tmp = fma((0.5 * x), (x * (eps * eps)), 1.0);
	} else {
		tmp = (x * x) * (0.5 * (eps * eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.6)
		tmp = fma(Float64(0.5 * x), Float64(x * Float64(eps * eps)), 1.0);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 * Float64(eps * eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.6], N[(N[(0.5 * x), $MachinePrecision] * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 62.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 90.0

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

   8. Applied rewrites 90.0%

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

   if 2.60000000000000009 < 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 42.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.6

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

   8. Applied rewrites 57.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, x \cdot \left(\varepsilon \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.7% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.49:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \varepsilon, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.49) (* eps (* x -0.5)) (fma x (fma 0.5 eps -0.5) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.49) {
		tmp = eps * (x * -0.5);
	} else {
		tmp = fma(x, fma(0.5, eps, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.49)
		tmp = Float64(eps * Float64(x * -0.5));
	else
		tmp = fma(x, fma(0.5, eps, -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.49], N[(eps * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 * eps + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.48999999999999999

   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

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.7

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

   8. Applied rewrites 24.7%

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

   if -0.48999999999999999 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.0%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. sub-neg N/A

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

       7. metadata-eval N/A

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 58.2

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

   11. Applied rewrites 58.2%

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

Alternative 15: 47.1% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.49:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x -0.49) (* eps (* x -0.5)) 1.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.49) {
		tmp = eps * (x * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.49d0)) then
        tmp = eps * (x * (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.49) {
		tmp = eps * (x * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.49:
		tmp = eps * (x * -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.49)
		tmp = Float64(eps * Float64(x * -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.49)
		tmp = eps * (x * -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.49], N[(eps * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.48999999999999999

   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

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.7

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

   8. Applied rewrites 24.7%

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

   if -0.48999999999999999 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.9% 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 72.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.0%

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

Reproduce

herbie shell --seed 2024212 
(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))