NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.8% → 99.8%

Time: 16.0s

Alternatives: 12

Speedup: 9.7×

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

• could not determine a ground truth

  1. x = -1.349808578186668e+38
  2. eps = -4.610311818900096e-249

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := \frac{1 + \varepsilon}{\varepsilon}\\ \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + t\_0 \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;0.5 \cdot \frac{2 + \left(x + x\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-1}\right)}^{\left(-\log \left(0.5 \cdot \mathsf{fma}\left(t\_0, t\_1, t\_1 \cdot e^{\varepsilon \cdot x - x}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps)))) (t_1 (/ (+ 1.0 eps) eps)))
   (if (<=
        (+
         (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
         (* t_0 (+ 1.0 (/ -1.0 eps))))
        2.0)
     (* 0.5 (/ (+ 2.0 (+ x x)) (exp x)))
     (pow
      (exp -1.0)
      (- (log (* 0.5 (fma t_0 t_1 (* t_1 (exp (- (* eps x) x)))))))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double t_1 = (1.0 + eps) / eps;
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps)))) <= 2.0) {
		tmp = 0.5 * ((2.0 + (x + x)) / exp(x));
	} else {
		tmp = pow(exp(-1.0), -log((0.5 * fma(t_0, t_1, (t_1 * exp(((eps * x) - x)))))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(-1.0 - eps)))
	t_1 = Float64(Float64(1.0 + eps) / eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(t_0 * Float64(1.0 + Float64(-1.0 / eps)))) <= 2.0)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(x + x)) / exp(x)));
	else
		tmp = exp(-1.0) ^ Float64(-log(Float64(0.5 * fma(t_0, t_1, Float64(t_1 * exp(Float64(Float64(eps * x) - x)))))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(0.5 * N[(N[(2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Exp[-1.0], $MachinePrecision], (-N[Log[N[(0.5 * N[(t$95$0 * t$95$1 + N[(t$95$1 * N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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))))) < 2

   1. Initial program 55.7%

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

   3. Taylor expanded in eps around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

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

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := 1 + \frac{-1}{\varepsilon}\\ t_2 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)}\\ \mathbf{if}\;t\_2 + t\_0 \cdot t\_1 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{2 + \left(x + x\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\log \left(\frac{2}{\mathsf{fma}\left(t\_0, t\_1, t\_2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps))))
        (t_1 (+ 1.0 (/ -1.0 eps)))
        (t_2 (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))))
   (if (<= (+ t_2 (* t_0 t_1)) 0.0)
     (* 0.5 (/ (+ 2.0 (+ x x)) (exp x)))
     (exp (- (log (/ 2.0 (fma t_0 t_1 t_2))))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double t_1 = 1.0 + (-1.0 / eps);
	double t_2 = (1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)));
	double tmp;
	if ((t_2 + (t_0 * t_1)) <= 0.0) {
		tmp = 0.5 * ((2.0 + (x + x)) / exp(x));
	} else {
		tmp = exp(-log((2.0 / fma(t_0, t_1, t_2))));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(-1.0 - eps)))
	t_1 = Float64(1.0 + Float64(-1.0 / eps))
	t_2 = Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0))))
	tmp = 0.0
	if (Float64(t_2 + Float64(t_0 * t_1)) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(x + x)) / exp(x)));
	else
		tmp = exp(Float64(-log(Float64(2.0 / fma(t_0, t_1, t_2)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[(-N[Log[N[(2.0 / N[(t$95$0 * t$95$1 + t$95$2), $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 42.8%

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

   3. Taylor expanded in eps around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   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 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.8%

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

   3. Taylor expanded in eps around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   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 99.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternative 4: 92.3% accurate, 0.7× speedup

Math

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

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))))
      2.0)
   (* 0.5 (/ (+ 2.0 (+ x x)) (exp x)))
   (fma x (* x (fma 0.5 (* eps eps) -0.5)) 1.0)))

C

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

Julia

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

Wolfram

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

   1. Initial program 55.7%

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

   3. Taylor expanded in eps around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

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

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

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

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

      9. exp-lowering-exp.f64 100.0

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

   7. Applied egg-rr 100.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 88.8%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 88.8

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

   8. Simplified 88.8%

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

4. Final simplification 94.5%

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

Alternative 5: 92.3% accurate, 0.7× speedup

Math

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

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))))
      2.0)
   (* (exp (- x)) (+ 1.0 x))
   (fma x (* x (fma 0.5 (* eps eps) -0.5)) 1.0)))

C

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

Julia

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

Wolfram

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

   1. Initial program 55.7%

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

   3. Taylor expanded in eps around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

      10. neg-lowering-neg.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. neg-mul-1 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. rgt-mult-inverse N/A

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

      9. distribute-lft-out N/A

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

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

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

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

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

      12. neg-mul-1 N/A

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 100.0

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

   10. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 88.8%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 88.8

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

   8. Simplified 88.8%

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

4. Final simplification 94.5%

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

Alternative 6: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10^{+300}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

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))))
      1e+300)
   1.0
   (* (* eps eps) (* 0.5 (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300) {
		tmp = 1.0;
	} else {
		tmp = (eps * eps) * (0.5 * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 1d+300) then
        tmp = 1.0d0
    else
        tmp = (eps * eps) * (0.5d0 * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300) {
		tmp = 1.0;
	} else {
		tmp = (eps * eps) * (0.5 * (x * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300:
		tmp = 1.0
	else:
		tmp = (eps * eps) * (0.5 * (x * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 1e+300)
		tmp = 1.0;
	else
		tmp = Float64(Float64(eps * eps) * Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300)
		tmp = 1.0;
	else
		tmp = (eps * eps) * (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(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], 1e+300], 1.0, N[(N[(eps * eps), $MachinePrecision] * N[(0.5 * N[(x * x), $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))))) < 1.0000000000000001e300

   1. Initial program 56.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.9%

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

   if 1.0000000000000001e300 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 89.3

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

   8. Simplified 89.3%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 80.8

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

   11. Simplified 80.8%

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

   12. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. cube-mult N/A

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

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

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

       11. unpow2 N/A

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. associate-*l* N/A

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

       17. lft-mult-inverse N/A

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

       18. metadata-eval N/A

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

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

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

       20. unpow2 N/A

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

       21. *-lowering-*.f64 80.4

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

   14. Simplified 80.4%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10^{+300}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10^{+300}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      1e+300)
   1.0
   (* eps (* eps (* 0.5 (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300) {
		tmp = 1.0;
	} else {
		tmp = eps * (eps * (0.5 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 1d+300) then
        tmp = 1.0d0
    else
        tmp = eps * (eps * (0.5d0 * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300) {
		tmp = 1.0;
	} else {
		tmp = eps * (eps * (0.5 * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300:
		tmp = 1.0
	else:
		tmp = eps * (eps * (0.5 * (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 1e+300)
		tmp = 1.0;
	else
		tmp = Float64(eps * Float64(eps * Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 1e+300)
		tmp = 1.0;
	else
		tmp = eps * (eps * (0.5 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(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], 1e+300], 1.0, N[(eps * N[(eps * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 1.0000000000000001e300

   1. Initial program 56.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 67.9%

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

   if 1.0000000000000001e300 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 89.3%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 89.3

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

   8. Simplified 89.3%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 72.3

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

   11. Simplified 72.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 10^{+300}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+159}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 550.0)
   1.0
   (if (<= x 1.6e+159) 0.0 (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 1.6e+159) {
		tmp = 0.0;
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 1.6e+159)
		tmp = 0.0;
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, If[LessEqual[x, 1.6e+159], 0.0, N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 550

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.5%

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

   if 550 < x < 1.59999999999999992e159

   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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 60.2

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

   7. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \]
   8. Step-by-step derivation

      1. div0 60.2

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

   9. Applied egg-rr 60.2%

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

   if 1.59999999999999992e159 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 N/A

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

   5. Simplified 37.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 64.2

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

   8. Simplified 64.2%

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

Alternative 9: 78.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.5, \varepsilon \cdot \varepsilon, -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 9.2e-6)
   (fma x (* x (fma 0.5 (* eps eps) -0.5)) 1.0)
   (* (* eps eps) (* 0.5 (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 9.2e-6) {
		tmp = fma(x, (x * fma(0.5, (eps * eps), -0.5)), 1.0);
	} else {
		tmp = (eps * eps) * (0.5 * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 9.2e-6)
		tmp = fma(x, Float64(x * fma(0.5, Float64(eps * eps), -0.5)), 1.0);
	else
		tmp = Float64(Float64(eps * eps) * Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 9.2e-6], N[(x * N[(x * N[(0.5 * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e-6

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 88.3%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 88.3

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

   8. Simplified 88.3%

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

   if 9.2e-6 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 48.4%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 47.9

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

   8. Simplified 47.9%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 48.3

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

   11. Simplified 48.3%

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

   12. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. cube-mult N/A

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

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

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

       11. unpow2 N/A

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. associate-*l* N/A

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

       17. lft-mult-inverse N/A

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

       18. metadata-eval N/A

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

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

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

       20. unpow2 N/A

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

       21. *-lowering-*.f64 63.4

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

   14. Simplified 63.4%

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

Alternative 10: 79.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 9.2e-6)
   (fma x (* eps (* eps (* x 0.5))) 1.0)
   (* (* eps eps) (* 0.5 (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 9.2e-6) {
		tmp = fma(x, (eps * (eps * (x * 0.5))), 1.0);
	} else {
		tmp = (eps * eps) * (0.5 * (x * x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 9.2e-6)
		tmp = fma(x, Float64(eps * Float64(eps * Float64(x * 0.5))), 1.0);
	else
		tmp = Float64(Float64(eps * eps) * Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 9.2e-6], N[(x * N[(eps * N[(eps * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e-6

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 88.3%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. *-lowering-*.f64 87.7

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

   8. Simplified 87.7%

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

   if 9.2e-6 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 48.4%

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

   6. Taylor expanded in eps around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 47.9

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

   8. Simplified 47.9%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 48.3

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

   11. Simplified 48.3%

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

   12. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. lft-mult-inverse N/A

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

       6. associate-*l* N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. cube-mult N/A

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

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

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

       11. unpow2 N/A

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. associate-*r* N/A

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

       16. associate-*l* N/A

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

       17. lft-mult-inverse N/A

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

       18. metadata-eval N/A

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

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

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

       20. unpow2 N/A

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

       21. *-lowering-*.f64 63.4

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

   14. Simplified 63.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, \varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot 0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.8% accurate, 38.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.5%

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

   if 550 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. distribute-rgt1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 48.5

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

   7. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \]
   8. Step-by-step derivation

      1. div0 48.5

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

   9. Applied egg-rr 48.5%

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

Alternative 12: 15.3% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 77.6%

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

4. Applied egg-rr 77.6%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 17.9

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

7. Simplified 17.9%

   \[\leadsto \color{blue}{\frac{0}{\varepsilon}} \]
8. Step-by-step derivation

   1. div0 17.9

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

9. Applied egg-rr 17.9%

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

Reproduce

herbie shell --seed 2024204 
(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))